COMPLEX routines for general (i.e., unsymmetric, in some cases rectangular) matrix
cgebak
USAGE:
info, v = NumRu::Lapack.cgebak( job, side, ilo, ihi, scale, v, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEBAK( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO )
* Purpose
* =======
*
* CGEBAK forms the right or left eigenvectors of a complex general
* matrix by backward transformation on the computed eigenvectors of the
* balanced matrix output by CGEBAL.
*
* Arguments
* =========
*
* JOB (input) CHARACTER*1
* Specifies the type of backward transformation required:
* = 'N', do nothing, return immediately;
* = 'P', do backward transformation for permutation only;
* = 'S', do backward transformation for scaling only;
* = 'B', do backward transformations for both permutation and
* scaling.
* JOB must be the same as the argument JOB supplied to CGEBAL.
*
* SIDE (input) CHARACTER*1
* = 'R': V contains right eigenvectors;
* = 'L': V contains left eigenvectors.
*
* N (input) INTEGER
* The number of rows of the matrix V. N >= 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* The integers ILO and IHI determined by CGEBAL.
* 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*
* SCALE (input) REAL array, dimension (N)
* Details of the permutation and scaling factors, as returned
* by CGEBAL.
*
* M (input) INTEGER
* The number of columns of the matrix V. M >= 0.
*
* V (input/output) COMPLEX array, dimension (LDV,M)
* On entry, the matrix of right or left eigenvectors to be
* transformed, as returned by CHSEIN or CTREVC.
* On exit, V is overwritten by the transformed eigenvectors.
*
* LDV (input) INTEGER
* The leading dimension of the array V. LDV >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* =====================================================================
*
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cgebal
USAGE:
ilo, ihi, scale, info, a = NumRu::Lapack.cgebal( job, a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
* Purpose
* =======
*
* CGEBAL balances a general complex matrix A. This involves, first,
* permuting A by a similarity transformation to isolate eigenvalues
* in the first 1 to ILO-1 and last IHI+1 to N elements on the
* diagonal; and second, applying a diagonal similarity transformation
* to rows and columns ILO to IHI to make the rows and columns as
* close in norm as possible. Both steps are optional.
*
* Balancing may reduce the 1-norm of the matrix, and improve the
* accuracy of the computed eigenvalues and/or eigenvectors.
*
* Arguments
* =========
*
* JOB (input) CHARACTER*1
* Specifies the operations to be performed on A:
* = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
* for i = 1,...,N;
* = 'P': permute only;
* = 'S': scale only;
* = 'B': both permute and scale.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the input matrix A.
* On exit, A is overwritten by the balanced matrix.
* If JOB = 'N', A is not referenced.
* See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* ILO (output) INTEGER
* IHI (output) INTEGER
* ILO and IHI are set to integers such that on exit
* A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
* If JOB = 'N' or 'S', ILO = 1 and IHI = N.
*
* SCALE (output) REAL array, dimension (N)
* Details of the permutations and scaling factors applied to
* A. If P(j) is the index of the row and column interchanged
* with row and column j and D(j) is the scaling factor
* applied to row and column j, then
* SCALE(j) = P(j) for j = 1,...,ILO-1
* = D(j) for j = ILO,...,IHI
* = P(j) for j = IHI+1,...,N.
* The order in which the interchanges are made is N to IHI+1,
* then 1 to ILO-1.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* The permutations consist of row and column interchanges which put
* the matrix in the form
*
* ( T1 X Y )
* P A P = ( 0 B Z )
* ( 0 0 T2 )
*
* where T1 and T2 are upper triangular matrices whose eigenvalues lie
* along the diagonal. The column indices ILO and IHI mark the starting
* and ending columns of the submatrix B. Balancing consists of applying
* a diagonal similarity transformation inv(D) * B * D to make the
* 1-norms of each row of B and its corresponding column nearly equal.
* The output matrix is
*
* ( T1 X*D Y )
* ( 0 inv(D)*B*D inv(D)*Z ).
* ( 0 0 T2 )
*
* Information about the permutations P and the diagonal matrix D is
* returned in the vector SCALE.
*
* This subroutine is based on the EISPACK routine CBAL.
*
* Modified by Tzu-Yi Chen, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
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cgebd2
USAGE:
d, e, tauq, taup, info, a = NumRu::Lapack.cgebd2( m, a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
* Purpose
* =======
*
* CGEBD2 reduces a complex general m by n matrix A to upper or lower
* real bidiagonal form B by a unitary transformation: Q' * A * P = B.
*
* If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows in the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns in the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the m by n general matrix to be reduced.
* On exit,
* if m >= n, the diagonal and the first superdiagonal are
* overwritten with the upper bidiagonal matrix B; the
* elements below the diagonal, with the array TAUQ, represent
* the unitary matrix Q as a product of elementary
* reflectors, and the elements above the first superdiagonal,
* with the array TAUP, represent the unitary matrix P as
* a product of elementary reflectors;
* if m < n, the diagonal and the first subdiagonal are
* overwritten with the lower bidiagonal matrix B; the
* elements below the first subdiagonal, with the array TAUQ,
* represent the unitary matrix Q as a product of
* elementary reflectors, and the elements above the diagonal,
* with the array TAUP, represent the unitary matrix P as
* a product of elementary reflectors.
* See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* D (output) REAL array, dimension (min(M,N))
* The diagonal elements of the bidiagonal matrix B:
* D(i) = A(i,i).
*
* E (output) REAL array, dimension (min(M,N)-1)
* The off-diagonal elements of the bidiagonal matrix B:
* if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
* if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
*
* TAUQ (output) COMPLEX array dimension (min(M,N))
* The scalar factors of the elementary reflectors which
* represent the unitary matrix Q. See Further Details.
*
* TAUP (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors which
* represent the unitary matrix P. See Further Details.
*
* WORK (workspace) COMPLEX array, dimension (max(M,N))
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* The matrices Q and P are represented as products of elementary
* reflectors:
*
* If m >= n,
*
* Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
*
* Each H(i) and G(i) has the form:
*
* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
*
* where tauq and taup are complex scalars, and v and u are complex
* vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
* A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
* A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*
* If m < n,
*
* Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
*
* Each H(i) and G(i) has the form:
*
* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
*
* where tauq and taup are complex scalars, v and u are complex vectors;
* v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
* u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
* tauq is stored in TAUQ(i) and taup in TAUP(i).
*
* The contents of A on exit are illustrated by the following examples:
*
* m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*
* ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
* ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
* ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
* ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
* ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
* ( v1 v2 v3 v4 v5 )
*
* where d and e denote diagonal and off-diagonal elements of B, vi
* denotes an element of the vector defining H(i), and ui an element of
* the vector defining G(i).
*
* =====================================================================
*
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cgebrd
USAGE:
d, e, tauq, taup, work, info, a = NumRu::Lapack.cgebrd( m, a, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO )
* Purpose
* =======
*
* CGEBRD reduces a general complex M-by-N matrix A to upper or lower
* bidiagonal form B by a unitary transformation: Q**H * A * P = B.
*
* If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows in the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns in the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N general matrix to be reduced.
* On exit,
* if m >= n, the diagonal and the first superdiagonal are
* overwritten with the upper bidiagonal matrix B; the
* elements below the diagonal, with the array TAUQ, represent
* the unitary matrix Q as a product of elementary
* reflectors, and the elements above the first superdiagonal,
* with the array TAUP, represent the unitary matrix P as
* a product of elementary reflectors;
* if m < n, the diagonal and the first subdiagonal are
* overwritten with the lower bidiagonal matrix B; the
* elements below the first subdiagonal, with the array TAUQ,
* represent the unitary matrix Q as a product of
* elementary reflectors, and the elements above the diagonal,
* with the array TAUP, represent the unitary matrix P as
* a product of elementary reflectors.
* See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* D (output) REAL array, dimension (min(M,N))
* The diagonal elements of the bidiagonal matrix B:
* D(i) = A(i,i).
*
* E (output) REAL array, dimension (min(M,N)-1)
* The off-diagonal elements of the bidiagonal matrix B:
* if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
* if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
*
* TAUQ (output) COMPLEX array dimension (min(M,N))
* The scalar factors of the elementary reflectors which
* represent the unitary matrix Q. See Further Details.
*
* TAUP (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors which
* represent the unitary matrix P. See Further Details.
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The length of the array WORK. LWORK >= max(1,M,N).
* For optimum performance LWORK >= (M+N)*NB, where NB
* is the optimal blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* The matrices Q and P are represented as products of elementary
* reflectors:
*
* If m >= n,
*
* Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
*
* Each H(i) and G(i) has the form:
*
* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
*
* where tauq and taup are complex scalars, and v and u are complex
* vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
* A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
* A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*
* If m < n,
*
* Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
*
* Each H(i) and G(i) has the form:
*
* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
*
* where tauq and taup are complex scalars, and v and u are complex
* vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
* A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
* A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*
* The contents of A on exit are illustrated by the following examples:
*
* m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*
* ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
* ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
* ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
* ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
* ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
* ( v1 v2 v3 v4 v5 )
*
* where d and e denote diagonal and off-diagonal elements of B, vi
* denotes an element of the vector defining H(i), and ui an element of
* the vector defining G(i).
*
* =====================================================================
*
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cgecon
USAGE:
rcond, info = NumRu::Lapack.cgecon( norm, a, anorm, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGECON( NORM, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO )
* Purpose
* =======
*
* CGECON estimates the reciprocal of the condition number of a general
* complex matrix A, in either the 1-norm or the infinity-norm, using
* the LU factorization computed by CGETRF.
*
* An estimate is obtained for norm(inv(A)), and the reciprocal of the
* condition number is computed as
* RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*
* Arguments
* =========
*
* NORM (input) CHARACTER*1
* Specifies whether the 1-norm condition number or the
* infinity-norm condition number is required:
* = '1' or 'O': 1-norm;
* = 'I': Infinity-norm.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input) COMPLEX array, dimension (LDA,N)
* The factors L and U from the factorization A = P*L*U
* as computed by CGETRF.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* ANORM (input) REAL
* If NORM = '1' or 'O', the 1-norm of the original matrix A.
* If NORM = 'I', the infinity-norm of the original matrix A.
*
* RCOND (output) REAL
* The reciprocal of the condition number of the matrix A,
* computed as RCOND = 1/(norm(A) * norm(inv(A))).
*
* WORK (workspace) COMPLEX array, dimension (2*N)
*
* RWORK (workspace) REAL array, dimension (2*N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
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cgeequ
USAGE:
r, c, rowcnd, colcnd, amax, info = NumRu::Lapack.cgeequ( a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEEQU( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO )
* Purpose
* =======
*
* CGEEQU computes row and column scalings intended to equilibrate an
* M-by-N matrix A and reduce its condition number. R returns the row
* scale factors and C the column scale factors, chosen to try to make
* the largest element in each row and column of the matrix B with
* elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
*
* R(i) and C(j) are restricted to be between SMLNUM = smallest safe
* number and BIGNUM = largest safe number. Use of these scaling
* factors is not guaranteed to reduce the condition number of A but
* works well in practice.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input) COMPLEX array, dimension (LDA,N)
* The M-by-N matrix whose equilibration factors are
* to be computed.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* R (output) REAL array, dimension (M)
* If INFO = 0 or INFO > M, R contains the row scale factors
* for A.
*
* C (output) REAL array, dimension (N)
* If INFO = 0, C contains the column scale factors for A.
*
* ROWCND (output) REAL
* If INFO = 0 or INFO > M, ROWCND contains the ratio of the
* smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
* AMAX is neither too large nor too small, it is not worth
* scaling by R.
*
* COLCND (output) REAL
* If INFO = 0, COLCND contains the ratio of the smallest
* C(i) to the largest C(i). If COLCND >= 0.1, it is not
* worth scaling by C.
*
* AMAX (output) REAL
* Absolute value of largest matrix element. If AMAX is very
* close to overflow or very close to underflow, the matrix
* should be scaled.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, and i is
* <= M: the i-th row of A is exactly zero
* > M: the (i-M)-th column of A is exactly zero
*
* =====================================================================
*
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cgeequb
USAGE:
r, c, rowcnd, colcnd, amax, info = NumRu::Lapack.cgeequb( a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO )
* Purpose
* =======
*
* CGEEQUB computes row and column scalings intended to equilibrate an
* M-by-N matrix A and reduce its condition number. R returns the row
* scale factors and C the column scale factors, chosen to try to make
* the largest element in each row and column of the matrix B with
* elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
* the radix.
*
* R(i) and C(j) are restricted to be a power of the radix between
* SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
* of these scaling factors is not guaranteed to reduce the condition
* number of A but works well in practice.
*
* This routine differs from CGEEQU by restricting the scaling factors
* to a power of the radix. Baring over- and underflow, scaling by
* these factors introduces no additional rounding errors. However, the
* scaled entries' magnitured are no longer approximately 1 but lie
* between sqrt(radix) and 1/sqrt(radix).
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input) COMPLEX array, dimension (LDA,N)
* The M-by-N matrix whose equilibration factors are
* to be computed.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* R (output) REAL array, dimension (M)
* If INFO = 0 or INFO > M, R contains the row scale factors
* for A.
*
* C (output) REAL array, dimension (N)
* If INFO = 0, C contains the column scale factors for A.
*
* ROWCND (output) REAL
* If INFO = 0 or INFO > M, ROWCND contains the ratio of the
* smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
* AMAX is neither too large nor too small, it is not worth
* scaling by R.
*
* COLCND (output) REAL
* If INFO = 0, COLCND contains the ratio of the smallest
* C(i) to the largest C(i). If COLCND >= 0.1, it is not
* worth scaling by C.
*
* AMAX (output) REAL
* Absolute value of largest matrix element. If AMAX is very
* close to overflow or very close to underflow, the matrix
* should be scaled.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, and i is
* <= M: the i-th row of A is exactly zero
* > M: the (i-M)-th column of A is exactly zero
*
* =====================================================================
*
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cgees
USAGE:
sdim, w, vs, work, info, a = NumRu::Lapack.cgees( jobvs, sort, a, [:lwork => lwork, :usage => usage, :help => help]){|a| ... }
FORTRAN MANUAL
SUBROUTINE CGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, W, VS, LDVS, WORK, LWORK, RWORK, BWORK, INFO )
* Purpose
* =======
*
* CGEES computes for an N-by-N complex nonsymmetric matrix A, the
* eigenvalues, the Schur form T, and, optionally, the matrix of Schur
* vectors Z. This gives the Schur factorization A = Z*T*(Z**H).
*
* Optionally, it also orders the eigenvalues on the diagonal of the
* Schur form so that selected eigenvalues are at the top left.
* The leading columns of Z then form an orthonormal basis for the
* invariant subspace corresponding to the selected eigenvalues.
* A complex matrix is in Schur form if it is upper triangular.
*
* Arguments
* =========
*
* JOBVS (input) CHARACTER*1
* = 'N': Schur vectors are not computed;
* = 'V': Schur vectors are computed.
*
* SORT (input) CHARACTER*1
* Specifies whether or not to order the eigenvalues on the
* diagonal of the Schur form.
* = 'N': Eigenvalues are not ordered:
* = 'S': Eigenvalues are ordered (see SELECT).
*
* SELECT (external procedure) LOGICAL FUNCTION of one COMPLEX argument
* SELECT must be declared EXTERNAL in the calling subroutine.
* If SORT = 'S', SELECT is used to select eigenvalues to order
* to the top left of the Schur form.
* IF SORT = 'N', SELECT is not referenced.
* The eigenvalue W(j) is selected if SELECT(W(j)) is true.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the N-by-N matrix A.
* On exit, A has been overwritten by its Schur form T.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* SDIM (output) INTEGER
* If SORT = 'N', SDIM = 0.
* If SORT = 'S', SDIM = number of eigenvalues for which
* SELECT is true.
*
* W (output) COMPLEX array, dimension (N)
* W contains the computed eigenvalues, in the same order that
* they appear on the diagonal of the output Schur form T.
*
* VS (output) COMPLEX array, dimension (LDVS,N)
* If JOBVS = 'V', VS contains the unitary matrix Z of Schur
* vectors.
* If JOBVS = 'N', VS is not referenced.
*
* LDVS (input) INTEGER
* The leading dimension of the array VS. LDVS >= 1; if
* JOBVS = 'V', LDVS >= N.
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,2*N).
* For good performance, LWORK must generally be larger.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* RWORK (workspace) REAL array, dimension (N)
*
* BWORK (workspace) LOGICAL array, dimension (N)
* Not referenced if SORT = 'N'.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = i, and i is
* <= N: the QR algorithm failed to compute all the
* eigenvalues; elements 1:ILO-1 and i+1:N of W
* contain those eigenvalues which have converged;
* if JOBVS = 'V', VS contains the matrix which
* reduces A to its partially converged Schur form.
* = N+1: the eigenvalues could not be reordered because
* some eigenvalues were too close to separate (the
* problem is very ill-conditioned);
* = N+2: after reordering, roundoff changed values of
* some complex eigenvalues so that leading
* eigenvalues in the Schur form no longer satisfy
* SELECT = .TRUE.. This could also be caused by
* underflow due to scaling.
*
* =====================================================================
*
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cgeesx
USAGE:
sdim, w, vs, rconde, rcondv, work, info, a = NumRu::Lapack.cgeesx( jobvs, sort, sense, a, [:lwork => lwork, :usage => usage, :help => help]){|a| ... }
FORTRAN MANUAL
SUBROUTINE CGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK, BWORK, INFO )
* Purpose
* =======
*
* CGEESX computes for an N-by-N complex nonsymmetric matrix A, the
* eigenvalues, the Schur form T, and, optionally, the matrix of Schur
* vectors Z. This gives the Schur factorization A = Z*T*(Z**H).
*
* Optionally, it also orders the eigenvalues on the diagonal of the
* Schur form so that selected eigenvalues are at the top left;
* computes a reciprocal condition number for the average of the
* selected eigenvalues (RCONDE); and computes a reciprocal condition
* number for the right invariant subspace corresponding to the
* selected eigenvalues (RCONDV). The leading columns of Z form an
* orthonormal basis for this invariant subspace.
*
* For further explanation of the reciprocal condition numbers RCONDE
* and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
* these quantities are called s and sep respectively).
*
* A complex matrix is in Schur form if it is upper triangular.
*
* Arguments
* =========
*
* JOBVS (input) CHARACTER*1
* = 'N': Schur vectors are not computed;
* = 'V': Schur vectors are computed.
*
* SORT (input) CHARACTER*1
* Specifies whether or not to order the eigenvalues on the
* diagonal of the Schur form.
* = 'N': Eigenvalues are not ordered;
* = 'S': Eigenvalues are ordered (see SELECT).
*
* SELECT (external procedure) LOGICAL FUNCTION of one COMPLEX argument
* SELECT must be declared EXTERNAL in the calling subroutine.
* If SORT = 'S', SELECT is used to select eigenvalues to order
* to the top left of the Schur form.
* If SORT = 'N', SELECT is not referenced.
* An eigenvalue W(j) is selected if SELECT(W(j)) is true.
*
* SENSE (input) CHARACTER*1
* Determines which reciprocal condition numbers are computed.
* = 'N': None are computed;
* = 'E': Computed for average of selected eigenvalues only;
* = 'V': Computed for selected right invariant subspace only;
* = 'B': Computed for both.
* If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA, N)
* On entry, the N-by-N matrix A.
* On exit, A is overwritten by its Schur form T.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* SDIM (output) INTEGER
* If SORT = 'N', SDIM = 0.
* If SORT = 'S', SDIM = number of eigenvalues for which
* SELECT is true.
*
* W (output) COMPLEX array, dimension (N)
* W contains the computed eigenvalues, in the same order
* that they appear on the diagonal of the output Schur form T.
*
* VS (output) COMPLEX array, dimension (LDVS,N)
* If JOBVS = 'V', VS contains the unitary matrix Z of Schur
* vectors.
* If JOBVS = 'N', VS is not referenced.
*
* LDVS (input) INTEGER
* The leading dimension of the array VS. LDVS >= 1, and if
* JOBVS = 'V', LDVS >= N.
*
* RCONDE (output) REAL
* If SENSE = 'E' or 'B', RCONDE contains the reciprocal
* condition number for the average of the selected eigenvalues.
* Not referenced if SENSE = 'N' or 'V'.
*
* RCONDV (output) REAL
* If SENSE = 'V' or 'B', RCONDV contains the reciprocal
* condition number for the selected right invariant subspace.
* Not referenced if SENSE = 'N' or 'E'.
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,2*N).
* Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
* where SDIM is the number of selected eigenvalues computed by
* this routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
* that an error is only returned if LWORK < max(1,2*N), but if
* SENSE = 'E' or 'V' or 'B' this may not be large enough.
* For good performance, LWORK must generally be larger.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates upper bound on the optimal size of the
* array WORK, returns this value as the first entry of the WORK
* array, and no error message related to LWORK is issued by
* XERBLA.
*
* RWORK (workspace) REAL array, dimension (N)
*
* BWORK (workspace) LOGICAL array, dimension (N)
* Not referenced if SORT = 'N'.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = i, and i is
* <= N: the QR algorithm failed to compute all the
* eigenvalues; elements 1:ILO-1 and i+1:N of W
* contain those eigenvalues which have converged; if
* JOBVS = 'V', VS contains the transformation which
* reduces A to its partially converged Schur form.
* = N+1: the eigenvalues could not be reordered because some
* eigenvalues were too close to separate (the problem
* is very ill-conditioned);
* = N+2: after reordering, roundoff changed values of some
* complex eigenvalues so that leading eigenvalues in
* the Schur form no longer satisfy SELECT=.TRUE. This
* could also be caused by underflow due to scaling.
*
* =====================================================================
*
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cgeev
USAGE:
w, vl, vr, work, info, a = NumRu::Lapack.cgeev( jobvl, jobvr, a, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEEV( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
* Purpose
* =======
*
* CGEEV computes for an N-by-N complex nonsymmetric matrix A, the
* eigenvalues and, optionally, the left and/or right eigenvectors.
*
* The right eigenvector v(j) of A satisfies
* A * v(j) = lambda(j) * v(j)
* where lambda(j) is its eigenvalue.
* The left eigenvector u(j) of A satisfies
* u(j)**H * A = lambda(j) * u(j)**H
* where u(j)**H denotes the conjugate transpose of u(j).
*
* The computed eigenvectors are normalized to have Euclidean norm
* equal to 1 and largest component real.
*
* Arguments
* =========
*
* JOBVL (input) CHARACTER*1
* = 'N': left eigenvectors of A are not computed;
* = 'V': left eigenvectors of are computed.
*
* JOBVR (input) CHARACTER*1
* = 'N': right eigenvectors of A are not computed;
* = 'V': right eigenvectors of A are computed.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the N-by-N matrix A.
* On exit, A has been overwritten.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* W (output) COMPLEX array, dimension (N)
* W contains the computed eigenvalues.
*
* VL (output) COMPLEX array, dimension (LDVL,N)
* If JOBVL = 'V', the left eigenvectors u(j) are stored one
* after another in the columns of VL, in the same order
* as their eigenvalues.
* If JOBVL = 'N', VL is not referenced.
* u(j) = VL(:,j), the j-th column of VL.
*
* LDVL (input) INTEGER
* The leading dimension of the array VL. LDVL >= 1; if
* JOBVL = 'V', LDVL >= N.
*
* VR (output) COMPLEX array, dimension (LDVR,N)
* If JOBVR = 'V', the right eigenvectors v(j) are stored one
* after another in the columns of VR, in the same order
* as their eigenvalues.
* If JOBVR = 'N', VR is not referenced.
* v(j) = VR(:,j), the j-th column of VR.
*
* LDVR (input) INTEGER
* The leading dimension of the array VR. LDVR >= 1; if
* JOBVR = 'V', LDVR >= N.
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,2*N).
* For good performance, LWORK must generally be larger.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* RWORK (workspace) REAL array, dimension (2*N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = i, the QR algorithm failed to compute all the
* eigenvalues, and no eigenvectors have been computed;
* elements and i+1:N of W contain eigenvalues which have
* converged.
*
* =====================================================================
*
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cgeevx
USAGE:
w, vl, vr, ilo, ihi, scale, abnrm, rconde, rcondv, work, info, a = NumRu::Lapack.cgeevx( balanc, jobvl, jobvr, sense, a, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, INFO )
* Purpose
* =======
*
* CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
* eigenvalues and, optionally, the left and/or right eigenvectors.
*
* Optionally also, it computes a balancing transformation to improve
* the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
* SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
* (RCONDE), and reciprocal condition numbers for the right
* eigenvectors (RCONDV).
*
* The right eigenvector v(j) of A satisfies
* A * v(j) = lambda(j) * v(j)
* where lambda(j) is its eigenvalue.
* The left eigenvector u(j) of A satisfies
* u(j)**H * A = lambda(j) * u(j)**H
* where u(j)**H denotes the conjugate transpose of u(j).
*
* The computed eigenvectors are normalized to have Euclidean norm
* equal to 1 and largest component real.
*
* Balancing a matrix means permuting the rows and columns to make it
* more nearly upper triangular, and applying a diagonal similarity
* transformation D * A * D**(-1), where D is a diagonal matrix, to
* make its rows and columns closer in norm and the condition numbers
* of its eigenvalues and eigenvectors smaller. The computed
* reciprocal condition numbers correspond to the balanced matrix.
* Permuting rows and columns will not change the condition numbers
* (in exact arithmetic) but diagonal scaling will. For further
* explanation of balancing, see section 4.10.2 of the LAPACK
* Users' Guide.
*
* Arguments
* =========
*
* BALANC (input) CHARACTER*1
* Indicates how the input matrix should be diagonally scaled
* and/or permuted to improve the conditioning of its
* eigenvalues.
* = 'N': Do not diagonally scale or permute;
* = 'P': Perform permutations to make the matrix more nearly
* upper triangular. Do not diagonally scale;
* = 'S': Diagonally scale the matrix, ie. replace A by
* D*A*D**(-1), where D is a diagonal matrix chosen
* to make the rows and columns of A more equal in
* norm. Do not permute;
* = 'B': Both diagonally scale and permute A.
*
* Computed reciprocal condition numbers will be for the matrix
* after balancing and/or permuting. Permuting does not change
* condition numbers (in exact arithmetic), but balancing does.
*
* JOBVL (input) CHARACTER*1
* = 'N': left eigenvectors of A are not computed;
* = 'V': left eigenvectors of A are computed.
* If SENSE = 'E' or 'B', JOBVL must = 'V'.
*
* JOBVR (input) CHARACTER*1
* = 'N': right eigenvectors of A are not computed;
* = 'V': right eigenvectors of A are computed.
* If SENSE = 'E' or 'B', JOBVR must = 'V'.
*
* SENSE (input) CHARACTER*1
* Determines which reciprocal condition numbers are computed.
* = 'N': None are computed;
* = 'E': Computed for eigenvalues only;
* = 'V': Computed for right eigenvectors only;
* = 'B': Computed for eigenvalues and right eigenvectors.
*
* If SENSE = 'E' or 'B', both left and right eigenvectors
* must also be computed (JOBVL = 'V' and JOBVR = 'V').
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the N-by-N matrix A.
* On exit, A has been overwritten. If JOBVL = 'V' or
* JOBVR = 'V', A contains the Schur form of the balanced
* version of the matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* W (output) COMPLEX array, dimension (N)
* W contains the computed eigenvalues.
*
* VL (output) COMPLEX array, dimension (LDVL,N)
* If JOBVL = 'V', the left eigenvectors u(j) are stored one
* after another in the columns of VL, in the same order
* as their eigenvalues.
* If JOBVL = 'N', VL is not referenced.
* u(j) = VL(:,j), the j-th column of VL.
*
* LDVL (input) INTEGER
* The leading dimension of the array VL. LDVL >= 1; if
* JOBVL = 'V', LDVL >= N.
*
* VR (output) COMPLEX array, dimension (LDVR,N)
* If JOBVR = 'V', the right eigenvectors v(j) are stored one
* after another in the columns of VR, in the same order
* as their eigenvalues.
* If JOBVR = 'N', VR is not referenced.
* v(j) = VR(:,j), the j-th column of VR.
*
* LDVR (input) INTEGER
* The leading dimension of the array VR. LDVR >= 1; if
* JOBVR = 'V', LDVR >= N.
*
* ILO (output) INTEGER
* IHI (output) INTEGER
* ILO and IHI are integer values determined when A was
* balanced. The balanced A(i,j) = 0 if I > J and
* J = 1,...,ILO-1 or I = IHI+1,...,N.
*
* SCALE (output) REAL array, dimension (N)
* Details of the permutations and scaling factors applied
* when balancing A. If P(j) is the index of the row and column
* interchanged with row and column j, and D(j) is the scaling
* factor applied to row and column j, then
* SCALE(J) = P(J), for J = 1,...,ILO-1
* = D(J), for J = ILO,...,IHI
* = P(J) for J = IHI+1,...,N.
* The order in which the interchanges are made is N to IHI+1,
* then 1 to ILO-1.
*
* ABNRM (output) REAL
* The one-norm of the balanced matrix (the maximum
* of the sum of absolute values of elements of any column).
*
* RCONDE (output) REAL array, dimension (N)
* RCONDE(j) is the reciprocal condition number of the j-th
* eigenvalue.
*
* RCONDV (output) REAL array, dimension (N)
* RCONDV(j) is the reciprocal condition number of the j-th
* right eigenvector.
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. If SENSE = 'N' or 'E',
* LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
* LWORK >= N*N+2*N.
* For good performance, LWORK must generally be larger.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* RWORK (workspace) REAL array, dimension (2*N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = i, the QR algorithm failed to compute all the
* eigenvalues, and no eigenvectors or condition numbers
* have been computed; elements 1:ILO-1 and i+1:N of W
* contain eigenvalues which have converged.
*
* =====================================================================
*
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cgegs
USAGE:
alpha, beta, vsl, vsr, work, info, a, b = NumRu::Lapack.cgegs( jobvsl, jobvsr, a, b, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, INFO )
* Purpose
* =======
*
* This routine is deprecated and has been replaced by routine CGGES.
*
* CGEGS computes the eigenvalues, Schur form, and, optionally, the
* left and or/right Schur vectors of a complex matrix pair (A,B).
* Given two square matrices A and B, the generalized Schur
* factorization has the form
*
* A = Q*S*Z**H, B = Q*T*Z**H
*
* where Q and Z are unitary matrices and S and T are upper triangular.
* The columns of Q are the left Schur vectors
* and the columns of Z are the right Schur vectors.
*
* If only the eigenvalues of (A,B) are needed, the driver routine
* CGEGV should be used instead. See CGEGV for a description of the
* eigenvalues of the generalized nonsymmetric eigenvalue problem
* (GNEP).
*
* Arguments
* =========
*
* JOBVSL (input) CHARACTER*1
* = 'N': do not compute the left Schur vectors;
* = 'V': compute the left Schur vectors (returned in VSL).
*
* JOBVSR (input) CHARACTER*1
* = 'N': do not compute the right Schur vectors;
* = 'V': compute the right Schur vectors (returned in VSR).
*
* N (input) INTEGER
* The order of the matrices A, B, VSL, and VSR. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA, N)
* On entry, the matrix A.
* On exit, the upper triangular matrix S from the generalized
* Schur factorization.
*
* LDA (input) INTEGER
* The leading dimension of A. LDA >= max(1,N).
*
* B (input/output) COMPLEX array, dimension (LDB, N)
* On entry, the matrix B.
* On exit, the upper triangular matrix T from the generalized
* Schur factorization.
*
* LDB (input) INTEGER
* The leading dimension of B. LDB >= max(1,N).
*
* ALPHA (output) COMPLEX array, dimension (N)
* The complex scalars alpha that define the eigenvalues of
* GNEP. ALPHA(j) = S(j,j), the diagonal element of the Schur
* form of A.
*
* BETA (output) COMPLEX array, dimension (N)
* The non-negative real scalars beta that define the
* eigenvalues of GNEP. BETA(j) = T(j,j), the diagonal element
* of the triangular factor T.
*
* Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
* represent the j-th eigenvalue of the matrix pair (A,B), in
* one of the forms lambda = alpha/beta or mu = beta/alpha.
* Since either lambda or mu may overflow, they should not,
* in general, be computed.
*
* VSL (output) COMPLEX array, dimension (LDVSL,N)
* If JOBVSL = 'V', the matrix of left Schur vectors Q.
* Not referenced if JOBVSL = 'N'.
*
* LDVSL (input) INTEGER
* The leading dimension of the matrix VSL. LDVSL >= 1, and
* if JOBVSL = 'V', LDVSL >= N.
*
* VSR (output) COMPLEX array, dimension (LDVSR,N)
* If JOBVSR = 'V', the matrix of right Schur vectors Z.
* Not referenced if JOBVSR = 'N'.
*
* LDVSR (input) INTEGER
* The leading dimension of the matrix VSR. LDVSR >= 1, and
* if JOBVSR = 'V', LDVSR >= N.
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,2*N).
* For good performance, LWORK must generally be larger.
* To compute the optimal value of LWORK, call ILAENV to get
* blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute:
* NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR;
* the optimal LWORK is N*(NB+1).
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* RWORK (workspace) REAL array, dimension (3*N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* =1,...,N:
* The QZ iteration failed. (A,B) are not in Schur
* form, but ALPHA(j) and BETA(j) should be correct for
* j=INFO+1,...,N.
* > N: errors that usually indicate LAPACK problems:
* =N+1: error return from CGGBAL
* =N+2: error return from CGEQRF
* =N+3: error return from CUNMQR
* =N+4: error return from CUNGQR
* =N+5: error return from CGGHRD
* =N+6: error return from CHGEQZ (other than failed
* iteration)
* =N+7: error return from CGGBAK (computing VSL)
* =N+8: error return from CGGBAK (computing VSR)
* =N+9: error return from CLASCL (various places)
*
* =====================================================================
*
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cgegv
USAGE:
alpha, beta, vl, vr, work, rwork, info, a, b = NumRu::Lapack.cgegv( jobvl, jobvr, a, b, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
* Purpose
* =======
*
* This routine is deprecated and has been replaced by routine CGGEV.
*
* CGEGV computes the eigenvalues and, optionally, the left and/or right
* eigenvectors of a complex matrix pair (A,B).
* Given two square matrices A and B,
* the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
* eigenvalues lambda and corresponding (non-zero) eigenvectors x such
* that
* A*x = lambda*B*x.
*
* An alternate form is to find the eigenvalues mu and corresponding
* eigenvectors y such that
* mu*A*y = B*y.
*
* These two forms are equivalent with mu = 1/lambda and x = y if
* neither lambda nor mu is zero. In order to deal with the case that
* lambda or mu is zero or small, two values alpha and beta are returned
* for each eigenvalue, such that lambda = alpha/beta and
* mu = beta/alpha.
*
* The vectors x and y in the above equations are right eigenvectors of
* the matrix pair (A,B). Vectors u and v satisfying
* u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
* are left eigenvectors of (A,B).
*
* Note: this routine performs "full balancing" on A and B -- see
* "Further Details", below.
*
* Arguments
* =========
*
* JOBVL (input) CHARACTER*1
* = 'N': do not compute the left generalized eigenvectors;
* = 'V': compute the left generalized eigenvectors (returned
* in VL).
*
* JOBVR (input) CHARACTER*1
* = 'N': do not compute the right generalized eigenvectors;
* = 'V': compute the right generalized eigenvectors (returned
* in VR).
*
* N (input) INTEGER
* The order of the matrices A, B, VL, and VR. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA, N)
* On entry, the matrix A.
* If JOBVL = 'V' or JOBVR = 'V', then on exit A
* contains the Schur form of A from the generalized Schur
* factorization of the pair (A,B) after balancing. If no
* eigenvectors were computed, then only the diagonal elements
* of the Schur form will be correct. See CGGHRD and CHGEQZ
* for details.
*
* LDA (input) INTEGER
* The leading dimension of A. LDA >= max(1,N).
*
* B (input/output) COMPLEX array, dimension (LDB, N)
* On entry, the matrix B.
* If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
* upper triangular matrix obtained from B in the generalized
* Schur factorization of the pair (A,B) after balancing.
* If no eigenvectors were computed, then only the diagonal
* elements of B will be correct. See CGGHRD and CHGEQZ for
* details.
*
* LDB (input) INTEGER
* The leading dimension of B. LDB >= max(1,N).
*
* ALPHA (output) COMPLEX array, dimension (N)
* The complex scalars alpha that define the eigenvalues of
* GNEP.
*
* BETA (output) COMPLEX array, dimension (N)
* The complex scalars beta that define the eigenvalues of GNEP.
*
* Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
* represent the j-th eigenvalue of the matrix pair (A,B), in
* one of the forms lambda = alpha/beta or mu = beta/alpha.
* Since either lambda or mu may overflow, they should not,
* in general, be computed.
*
* VL (output) COMPLEX array, dimension (LDVL,N)
* If JOBVL = 'V', the left eigenvectors u(j) are stored
* in the columns of VL, in the same order as their eigenvalues.
* Each eigenvector is scaled so that its largest component has
* abs(real part) + abs(imag. part) = 1, except for eigenvectors
* corresponding to an eigenvalue with alpha = beta = 0, which
* are set to zero.
* Not referenced if JOBVL = 'N'.
*
* LDVL (input) INTEGER
* The leading dimension of the matrix VL. LDVL >= 1, and
* if JOBVL = 'V', LDVL >= N.
*
* VR (output) COMPLEX array, dimension (LDVR,N)
* If JOBVR = 'V', the right eigenvectors x(j) are stored
* in the columns of VR, in the same order as their eigenvalues.
* Each eigenvector is scaled so that its largest component has
* abs(real part) + abs(imag. part) = 1, except for eigenvectors
* corresponding to an eigenvalue with alpha = beta = 0, which
* are set to zero.
* Not referenced if JOBVR = 'N'.
*
* LDVR (input) INTEGER
* The leading dimension of the matrix VR. LDVR >= 1, and
* if JOBVR = 'V', LDVR >= N.
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,2*N).
* For good performance, LWORK must generally be larger.
* To compute the optimal value of LWORK, call ILAENV to get
* blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute:
* NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR;
* The optimal LWORK is MAX( 2*N, N*(NB+1) ).
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* RWORK (workspace/output) REAL array, dimension (8*N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* =1,...,N:
* The QZ iteration failed. No eigenvectors have been
* calculated, but ALPHA(j) and BETA(j) should be
* correct for j=INFO+1,...,N.
* > N: errors that usually indicate LAPACK problems:
* =N+1: error return from CGGBAL
* =N+2: error return from CGEQRF
* =N+3: error return from CUNMQR
* =N+4: error return from CUNGQR
* =N+5: error return from CGGHRD
* =N+6: error return from CHGEQZ (other than failed
* iteration)
* =N+7: error return from CTGEVC
* =N+8: error return from CGGBAK (computing VL)
* =N+9: error return from CGGBAK (computing VR)
* =N+10: error return from CLASCL (various calls)
*
* Further Details
* ===============
*
* Balancing
* ---------
*
* This driver calls CGGBAL to both permute and scale rows and columns
* of A and B. The permutations PL and PR are chosen so that PL*A*PR
* and PL*B*R will be upper triangular except for the diagonal blocks
* A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
* possible. The diagonal scaling matrices DL and DR are chosen so
* that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
* one (except for the elements that start out zero.)
*
* After the eigenvalues and eigenvectors of the balanced matrices
* have been computed, CGGBAK transforms the eigenvectors back to what
* they would have been (in perfect arithmetic) if they had not been
* balanced.
*
* Contents of A and B on Exit
* -------- -- - --- - -- ----
*
* If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
* both), then on exit the arrays A and B will contain the complex Schur
* form[*] of the "balanced" versions of A and B. If no eigenvectors
* are computed, then only the diagonal blocks will be correct.
*
* [*] In other words, upper triangular form.
*
* =====================================================================
*
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cgehd2
USAGE:
tau, info, a = NumRu::Lapack.cgehd2( ilo, ihi, a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
* Purpose
* =======
*
* CGEHD2 reduces a complex general matrix A to upper Hessenberg form H
* by a unitary similarity transformation: Q' * A * Q = H .
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* It is assumed that A is already upper triangular in rows
* and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
* set by a previous call to CGEBAL; otherwise they should be
* set to 1 and N respectively. See Further Details.
* 1 <= ILO <= IHI <= max(1,N).
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the n by n general matrix to be reduced.
* On exit, the upper triangle and the first subdiagonal of A
* are overwritten with the upper Hessenberg matrix H, and the
* elements below the first subdiagonal, with the array TAU,
* represent the unitary matrix Q as a product of elementary
* reflectors. See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* TAU (output) COMPLEX array, dimension (N-1)
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace) COMPLEX array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of (ihi-ilo) elementary
* reflectors
*
* Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
* exit in A(i+2:ihi,i), and tau in TAU(i).
*
* The contents of A are illustrated by the following example, with
* n = 7, ilo = 2 and ihi = 6:
*
* on entry, on exit,
*
* ( a a a a a a a ) ( a a h h h h a )
* ( a a a a a a ) ( a h h h h a )
* ( a a a a a a ) ( h h h h h h )
* ( a a a a a a ) ( v2 h h h h h )
* ( a a a a a a ) ( v2 v3 h h h h )
* ( a a a a a a ) ( v2 v3 v4 h h h )
* ( a ) ( a )
*
* where a denotes an element of the original matrix A, h denotes a
* modified element of the upper Hessenberg matrix H, and vi denotes an
* element of the vector defining H(i).
*
* =====================================================================
*
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cgehrd
USAGE:
tau, work, info, a = NumRu::Lapack.cgehrd( ilo, ihi, a, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
* Purpose
* =======
*
* CGEHRD reduces a complex general matrix A to upper Hessenberg form H by
* an unitary similarity transformation: Q' * A * Q = H .
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* It is assumed that A is already upper triangular in rows
* and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
* set by a previous call to CGEBAL; otherwise they should be
* set to 1 and N respectively. See Further Details.
* 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the N-by-N general matrix to be reduced.
* On exit, the upper triangle and the first subdiagonal of A
* are overwritten with the upper Hessenberg matrix H, and the
* elements below the first subdiagonal, with the array TAU,
* represent the unitary matrix Q as a product of elementary
* reflectors. See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* TAU (output) COMPLEX array, dimension (N-1)
* The scalar factors of the elementary reflectors (see Further
* Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
* zero.
*
* WORK (workspace/output) COMPLEX array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The length of the array WORK. LWORK >= max(1,N).
* For optimum performance LWORK >= N*NB, where NB is the
* optimal blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of (ihi-ilo) elementary
* reflectors
*
* Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
* exit in A(i+2:ihi,i), and tau in TAU(i).
*
* The contents of A are illustrated by the following example, with
* n = 7, ilo = 2 and ihi = 6:
*
* on entry, on exit,
*
* ( a a a a a a a ) ( a a h h h h a )
* ( a a a a a a ) ( a h h h h a )
* ( a a a a a a ) ( h h h h h h )
* ( a a a a a a ) ( v2 h h h h h )
* ( a a a a a a ) ( v2 v3 h h h h )
* ( a a a a a a ) ( v2 v3 v4 h h h )
* ( a ) ( a )
*
* where a denotes an element of the original matrix A, h denotes a
* modified element of the upper Hessenberg matrix H, and vi denotes an
* element of the vector defining H(i).
*
* This file is a slight modification of LAPACK-3.0's DGEHRD
* subroutine incorporating improvements proposed by Quintana-Orti and
* Van de Geijn (2006). (See DLAHR2.)
*
* =====================================================================
*
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cgelq2
USAGE:
tau, info, a = NumRu::Lapack.cgelq2( a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGELQ2( M, N, A, LDA, TAU, WORK, INFO )
* Purpose
* =======
*
* CGELQ2 computes an LQ factorization of a complex m by n matrix A:
* A = L * Q.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the m by n matrix A.
* On exit, the elements on and below the diagonal of the array
* contain the m by min(m,n) lower trapezoidal matrix L (L is
* lower triangular if m <= n); the elements above the diagonal,
* with the array TAU, represent the unitary matrix Q as a
* product of elementary reflectors (see Further Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace) COMPLEX array, dimension (M)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(k)' . . . H(2)' H(1)', where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in
* A(i,i+1:n), and tau in TAU(i).
*
* =====================================================================
*
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cgelqf
USAGE:
tau, work, info, a = NumRu::Lapack.cgelqf( m, a, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
* Purpose
* =======
*
* CGELQF computes an LQ factorization of a complex M-by-N matrix A:
* A = L * Q.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, the elements on and below the diagonal of the array
* contain the m-by-min(m,n) lower trapezoidal matrix L (L is
* lower triangular if m <= n); the elements above the diagonal,
* with the array TAU, represent the unitary matrix Q as a
* product of elementary reflectors (see Further Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,M).
* For optimum performance LWORK >= M*NB, where NB is the
* optimal blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(k)' . . . H(2)' H(1)', where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in
* A(i,i+1:n), and tau in TAU(i).
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL CGELQ2, CLARFB, CLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
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cgels
USAGE:
work, info, a, b = NumRu::Lapack.cgels( trans, a, b, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO )
* Purpose
* =======
*
* CGELS solves overdetermined or underdetermined complex linear systems
* involving an M-by-N matrix A, or its conjugate-transpose, using a QR
* or LQ factorization of A. It is assumed that A has full rank.
*
* The following options are provided:
*
* 1. If TRANS = 'N' and m >= n: find the least squares solution of
* an overdetermined system, i.e., solve the least squares problem
* minimize || B - A*X ||.
*
* 2. If TRANS = 'N' and m < n: find the minimum norm solution of
* an underdetermined system A * X = B.
*
* 3. If TRANS = 'C' and m >= n: find the minimum norm solution of
* an undetermined system A**H * X = B.
*
* 4. If TRANS = 'C' and m < n: find the least squares solution of
* an overdetermined system, i.e., solve the least squares problem
* minimize || B - A**H * X ||.
*
* Several right hand side vectors b and solution vectors x can be
* handled in a single call; they are stored as the columns of the
* M-by-NRHS right hand side matrix B and the N-by-NRHS solution
* matrix X.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* = 'N': the linear system involves A;
* = 'C': the linear system involves A**H.
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of
* columns of the matrices B and X. NRHS >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* if M >= N, A is overwritten by details of its QR
* factorization as returned by CGEQRF;
* if M < N, A is overwritten by details of its LQ
* factorization as returned by CGELQF.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the matrix B of right hand side vectors, stored
* columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
* if TRANS = 'C'.
* On exit, if INFO = 0, B is overwritten by the solution
* vectors, stored columnwise:
* if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
* squares solution vectors; the residual sum of squares for the
* solution in each column is given by the sum of squares of the
* modulus of elements N+1 to M in that column;
* if TRANS = 'N' and m < n, rows 1 to N of B contain the
* minimum norm solution vectors;
* if TRANS = 'C' and m >= n, rows 1 to M of B contain the
* minimum norm solution vectors;
* if TRANS = 'C' and m < n, rows 1 to M of B contain the
* least squares solution vectors; the residual sum of squares
* for the solution in each column is given by the sum of
* squares of the modulus of elements M+1 to N in that column.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= MAX(1,M,N).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK.
* LWORK >= max( 1, MN + max( MN, NRHS ) ).
* For optimal performance,
* LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
* where MN = min(M,N) and NB is the optimum block size.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, the i-th diagonal element of the
* triangular factor of A is zero, so that A does not have
* full rank; the least squares solution could not be
* computed.
*
* =====================================================================
*
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cgelsd
USAGE:
s, rank, work, info, b = NumRu::Lapack.cgelsd( a, b, rcond, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, RWORK, IWORK, INFO )
* Purpose
* =======
*
* CGELSD computes the minimum-norm solution to a real linear least
* squares problem:
* minimize 2-norm(| b - A*x |)
* using the singular value decomposition (SVD) of A. A is an M-by-N
* matrix which may be rank-deficient.
*
* Several right hand side vectors b and solution vectors x can be
* handled in a single call; they are stored as the columns of the
* M-by-NRHS right hand side matrix B and the N-by-NRHS solution
* matrix X.
*
* The problem is solved in three steps:
* (1) Reduce the coefficient matrix A to bidiagonal form with
* Householder tranformations, reducing the original problem
* into a "bidiagonal least squares problem" (BLS)
* (2) Solve the BLS using a divide and conquer approach.
* (3) Apply back all the Householder tranformations to solve
* the original least squares problem.
*
* The effective rank of A is determined by treating as zero those
* singular values which are less than RCOND times the largest singular
* value.
*
* The divide and conquer algorithm makes very mild assumptions about
* floating point arithmetic. It will work on machines with a guard
* digit in add/subtract, or on those binary machines without guard
* digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
* Cray-2. It could conceivably fail on hexadecimal or decimal machines
* without guard digits, but we know of none.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, A has been destroyed.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the M-by-NRHS right hand side matrix B.
* On exit, B is overwritten by the N-by-NRHS solution matrix X.
* If m >= n and RANK = n, the residual sum-of-squares for
* the solution in the i-th column is given by the sum of
* squares of the modulus of elements n+1:m in that column.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,M,N).
*
* S (output) REAL array, dimension (min(M,N))
* The singular values of A in decreasing order.
* The condition number of A in the 2-norm = S(1)/S(min(m,n)).
*
* RCOND (input) REAL
* RCOND is used to determine the effective rank of A.
* Singular values S(i) <= RCOND*S(1) are treated as zero.
* If RCOND < 0, machine precision is used instead.
*
* RANK (output) INTEGER
* The effective rank of A, i.e., the number of singular values
* which are greater than RCOND*S(1).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK must be at least 1.
* The exact minimum amount of workspace needed depends on M,
* N and NRHS. As long as LWORK is at least
* 2 * N + N * NRHS
* if M is greater than or equal to N or
* 2 * M + M * NRHS
* if M is less than N, the code will execute correctly.
* For good performance, LWORK should generally be larger.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the array WORK and the
* minimum sizes of the arrays RWORK and IWORK, and returns
* these values as the first entries of the WORK, RWORK and
* IWORK arrays, and no error message related to LWORK is issued
* by XERBLA.
*
* RWORK (workspace) REAL array, dimension (MAX(1,LRWORK))
* LRWORK >=
* 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
* MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
* if M is greater than or equal to N or
* 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
* MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
* if M is less than N, the code will execute correctly.
* SMLSIZ is returned by ILAENV and is equal to the maximum
* size of the subproblems at the bottom of the computation
* tree (usually about 25), and
* NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
* On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.
*
* IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK))
* LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
* where MINMN = MIN( M,N ).
* On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: the algorithm for computing the SVD failed to converge;
* if INFO = i, i off-diagonal elements of an intermediate
* bidiagonal form did not converge to zero.
*
* Further Details
* ===============
*
* Based on contributions by
* Ming Gu and Ren-Cang Li, Computer Science Division, University of
* California at Berkeley, USA
* Osni Marques, LBNL/NERSC, USA
*
* =====================================================================
*
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cgelss
USAGE:
s, rank, work, info, a, b = NumRu::Lapack.cgelss( a, b, rcond, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, RWORK, INFO )
* Purpose
* =======
*
* CGELSS computes the minimum norm solution to a complex linear
* least squares problem:
*
* Minimize 2-norm(| b - A*x |).
*
* using the singular value decomposition (SVD) of A. A is an M-by-N
* matrix which may be rank-deficient.
*
* Several right hand side vectors b and solution vectors x can be
* handled in a single call; they are stored as the columns of the
* M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix
* X.
*
* The effective rank of A is determined by treating as zero those
* singular values which are less than RCOND times the largest singular
* value.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, the first min(m,n) rows of A are overwritten with
* its right singular vectors, stored rowwise.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the M-by-NRHS right hand side matrix B.
* On exit, B is overwritten by the N-by-NRHS solution matrix X.
* If m >= n and RANK = n, the residual sum-of-squares for
* the solution in the i-th column is given by the sum of
* squares of the modulus of elements n+1:m in that column.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,M,N).
*
* S (output) REAL array, dimension (min(M,N))
* The singular values of A in decreasing order.
* The condition number of A in the 2-norm = S(1)/S(min(m,n)).
*
* RCOND (input) REAL
* RCOND is used to determine the effective rank of A.
* Singular values S(i) <= RCOND*S(1) are treated as zero.
* If RCOND < 0, machine precision is used instead.
*
* RANK (output) INTEGER
* The effective rank of A, i.e., the number of singular values
* which are greater than RCOND*S(1).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= 1, and also:
* LWORK >= 2*min(M,N) + max(M,N,NRHS)
* For good performance, LWORK should generally be larger.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* RWORK (workspace) REAL array, dimension (5*min(M,N))
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: the algorithm for computing the SVD failed to converge;
* if INFO = i, i off-diagonal elements of an intermediate
* bidiagonal form did not converge to zero.
*
* =====================================================================
*
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cgelsx
USAGE:
rank, info, a, b, jpvt = NumRu::Lapack.cgelsx( m, a, b, jpvt, rcond, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, RWORK, INFO )
* Purpose
* =======
*
* This routine is deprecated and has been replaced by routine CGELSY.
*
* CGELSX computes the minimum-norm solution to a complex linear least
* squares problem:
* minimize || A * X - B ||
* using a complete orthogonal factorization of A. A is an M-by-N
* matrix which may be rank-deficient.
*
* Several right hand side vectors b and solution vectors x can be
* handled in a single call; they are stored as the columns of the
* M-by-NRHS right hand side matrix B and the N-by-NRHS solution
* matrix X.
*
* The routine first computes a QR factorization with column pivoting:
* A * P = Q * [ R11 R12 ]
* [ 0 R22 ]
* with R11 defined as the largest leading submatrix whose estimated
* condition number is less than 1/RCOND. The order of R11, RANK,
* is the effective rank of A.
*
* Then, R22 is considered to be negligible, and R12 is annihilated
* by unitary transformations from the right, arriving at the
* complete orthogonal factorization:
* A * P = Q * [ T11 0 ] * Z
* [ 0 0 ]
* The minimum-norm solution is then
* X = P * Z' [ inv(T11)*Q1'*B ]
* [ 0 ]
* where Q1 consists of the first RANK columns of Q.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of
* columns of matrices B and X. NRHS >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, A has been overwritten by details of its
* complete orthogonal factorization.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the M-by-NRHS right hand side matrix B.
* On exit, the N-by-NRHS solution matrix X.
* If m >= n and RANK = n, the residual sum-of-squares for
* the solution in the i-th column is given by the sum of
* squares of elements N+1:M in that column.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,M,N).
*
* JPVT (input/output) INTEGER array, dimension (N)
* On entry, if JPVT(i) .ne. 0, the i-th column of A is an
* initial column, otherwise it is a free column. Before
* the QR factorization of A, all initial columns are
* permuted to the leading positions; only the remaining
* free columns are moved as a result of column pivoting
* during the factorization.
* On exit, if JPVT(i) = k, then the i-th column of A*P
* was the k-th column of A.
*
* RCOND (input) REAL
* RCOND is used to determine the effective rank of A, which
* is defined as the order of the largest leading triangular
* submatrix R11 in the QR factorization with pivoting of A,
* whose estimated condition number < 1/RCOND.
*
* RANK (output) INTEGER
* The effective rank of A, i.e., the order of the submatrix
* R11. This is the same as the order of the submatrix T11
* in the complete orthogonal factorization of A.
*
* WORK (workspace) COMPLEX array, dimension
* (min(M,N) + max( N, 2*min(M,N)+NRHS )),
*
* RWORK (workspace) REAL array, dimension (2*N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
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cgelsy
USAGE:
rank, work, info, a, b, jpvt = NumRu::Lapack.cgelsy( a, b, jpvt, rcond, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, RWORK, INFO )
* Purpose
* =======
*
* CGELSY computes the minimum-norm solution to a complex linear least
* squares problem:
* minimize || A * X - B ||
* using a complete orthogonal factorization of A. A is an M-by-N
* matrix which may be rank-deficient.
*
* Several right hand side vectors b and solution vectors x can be
* handled in a single call; they are stored as the columns of the
* M-by-NRHS right hand side matrix B and the N-by-NRHS solution
* matrix X.
*
* The routine first computes a QR factorization with column pivoting:
* A * P = Q * [ R11 R12 ]
* [ 0 R22 ]
* with R11 defined as the largest leading submatrix whose estimated
* condition number is less than 1/RCOND. The order of R11, RANK,
* is the effective rank of A.
*
* Then, R22 is considered to be negligible, and R12 is annihilated
* by unitary transformations from the right, arriving at the
* complete orthogonal factorization:
* A * P = Q * [ T11 0 ] * Z
* [ 0 0 ]
* The minimum-norm solution is then
* X = P * Z' [ inv(T11)*Q1'*B ]
* [ 0 ]
* where Q1 consists of the first RANK columns of Q.
*
* This routine is basically identical to the original xGELSX except
* three differences:
* o The permutation of matrix B (the right hand side) is faster and
* more simple.
* o The call to the subroutine xGEQPF has been substituted by the
* the call to the subroutine xGEQP3. This subroutine is a Blas-3
* version of the QR factorization with column pivoting.
* o Matrix B (the right hand side) is updated with Blas-3.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of
* columns of matrices B and X. NRHS >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, A has been overwritten by details of its
* complete orthogonal factorization.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the M-by-NRHS right hand side matrix B.
* On exit, the N-by-NRHS solution matrix X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,M,N).
*
* JPVT (input/output) INTEGER array, dimension (N)
* On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
* to the front of AP, otherwise column i is a free column.
* On exit, if JPVT(i) = k, then the i-th column of A*P
* was the k-th column of A.
*
* RCOND (input) REAL
* RCOND is used to determine the effective rank of A, which
* is defined as the order of the largest leading triangular
* submatrix R11 in the QR factorization with pivoting of A,
* whose estimated condition number < 1/RCOND.
*
* RANK (output) INTEGER
* The effective rank of A, i.e., the order of the submatrix
* R11. This is the same as the order of the submatrix T11
* in the complete orthogonal factorization of A.
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK.
* The unblocked strategy requires that:
* LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
* where MN = min(M,N).
* The block algorithm requires that:
* LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
* where NB is an upper bound on the blocksize returned
* by ILAENV for the routines CGEQP3, CTZRZF, CTZRQF, CUNMQR,
* and CUNMRZ.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* RWORK (workspace) REAL array, dimension (2*N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* Based on contributions by
* A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
* E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
* G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
*
* =====================================================================
*
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cgeql2
USAGE:
tau, info, a = NumRu::Lapack.cgeql2( m, a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEQL2( M, N, A, LDA, TAU, WORK, INFO )
* Purpose
* =======
*
* CGEQL2 computes a QL factorization of a complex m by n matrix A:
* A = Q * L.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the m by n matrix A.
* On exit, if m >= n, the lower triangle of the subarray
* A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
* if m <= n, the elements on and below the (n-m)-th
* superdiagonal contain the m by n lower trapezoidal matrix L;
* the remaining elements, with the array TAU, represent the
* unitary matrix Q as a product of elementary reflectors
* (see Further Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace) COMPLEX array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(k) . . . H(2) H(1), where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
* A(1:m-k+i-1,n-k+i), and tau in TAU(i).
*
* =====================================================================
*
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cgeqlf
USAGE:
tau, work, info, a = NumRu::Lapack.cgeqlf( m, a, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEQLF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
* Purpose
* =======
*
* CGEQLF computes a QL factorization of a complex M-by-N matrix A:
* A = Q * L.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit,
* if m >= n, the lower triangle of the subarray
* A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
* if m <= n, the elements on and below the (n-m)-th
* superdiagonal contain the M-by-N lower trapezoidal matrix L;
* the remaining elements, with the array TAU, represent the
* unitary matrix Q as a product of elementary reflectors
* (see Further Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N).
* For optimum performance LWORK >= N*NB, where NB is
* the optimal blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(k) . . . H(2) H(1), where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
* A(1:m-k+i-1,n-k+i), and tau in TAU(i).
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
$ MU, NB, NBMIN, NU, NX
* ..
* .. External Subroutines ..
EXTERNAL CGEQL2, CLARFB, CLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
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cgeqp3
USAGE:
tau, work, info, a, jpvt = NumRu::Lapack.cgeqp3( m, a, jpvt, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, RWORK, INFO )
* Purpose
* =======
*
* CGEQP3 computes a QR factorization with column pivoting of a
* matrix A: A*P = Q*R using Level 3 BLAS.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, the upper triangle of the array contains the
* min(M,N)-by-N upper trapezoidal matrix R; the elements below
* the diagonal, together with the array TAU, represent the
* unitary matrix Q as a product of min(M,N) elementary
* reflectors.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* JPVT (input/output) INTEGER array, dimension (N)
* On entry, if JPVT(J).ne.0, the J-th column of A is permuted
* to the front of A*P (a leading column); if JPVT(J)=0,
* the J-th column of A is a free column.
* On exit, if JPVT(J)=K, then the J-th column of A*P was the
* the K-th column of A.
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors.
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO=0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= N+1.
* For optimal performance LWORK >= ( N+1 )*NB, where NB
* is the optimal blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* RWORK (workspace) REAL array, dimension (2*N)
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(1) H(2) . . . H(k), where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real/complex scalar, and v is a real/complex vector
* with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
* A(i+1:m,i), and tau in TAU(i).
*
* Based on contributions by
* G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
* X. Sun, Computer Science Dept., Duke University, USA
*
* =====================================================================
*
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cgeqpf
USAGE:
tau, info, a, jpvt = NumRu::Lapack.cgeqpf( m, a, jpvt, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEQPF( M, N, A, LDA, JPVT, TAU, WORK, RWORK, INFO )
* Purpose
* =======
*
* This routine is deprecated and has been replaced by routine CGEQP3.
*
* CGEQPF computes a QR factorization with column pivoting of a
* complex M-by-N matrix A: A*P = Q*R.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, the upper triangle of the array contains the
* min(M,N)-by-N upper triangular matrix R; the elements
* below the diagonal, together with the array TAU,
* represent the unitary matrix Q as a product of
* min(m,n) elementary reflectors.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* JPVT (input/output) INTEGER array, dimension (N)
* On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
* to the front of A*P (a leading column); if JPVT(i) = 0,
* the i-th column of A is a free column.
* On exit, if JPVT(i) = k, then the i-th column of A*P
* was the k-th column of A.
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors.
*
* WORK (workspace) COMPLEX array, dimension (N)
*
* RWORK (workspace) REAL array, dimension (2*N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(1) H(2) . . . H(n)
*
* Each H(i) has the form
*
* H = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).
*
* The matrix P is represented in jpvt as follows: If
* jpvt(j) = i
* then the jth column of P is the ith canonical unit vector.
*
* Partial column norm updating strategy modified by
* Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
* University of Zagreb, Croatia.
* June 2010
* For more details see LAPACK Working Note 176.
*
* =====================================================================
*
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cgeqr2
USAGE:
tau, info, a = NumRu::Lapack.cgeqr2( m, a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEQR2( M, N, A, LDA, TAU, WORK, INFO )
* Purpose
* =======
*
* CGEQR2 computes a QR factorization of a complex m by n matrix A:
* A = Q * R.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the m by n matrix A.
* On exit, the elements on and above the diagonal of the array
* contain the min(m,n) by n upper trapezoidal matrix R (R is
* upper triangular if m >= n); the elements below the diagonal,
* with the array TAU, represent the unitary matrix Q as a
* product of elementary reflectors (see Further Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace) COMPLEX array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(1) H(2) . . . H(k), where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
* and tau in TAU(i).
*
* =====================================================================
*
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cgeqr2p
USAGE:
tau, info, a = NumRu::Lapack.cgeqr2p( m, a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEQR2P( M, N, A, LDA, TAU, WORK, INFO )
* Purpose
* =======
*
* CGEQR2P computes a QR factorization of a complex m by n matrix A:
* A = Q * R.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the m by n matrix A.
* On exit, the elements on and above the diagonal of the array
* contain the min(m,n) by n upper trapezoidal matrix R (R is
* upper triangular if m >= n); the elements below the diagonal,
* with the array TAU, represent the unitary matrix Q as a
* product of elementary reflectors (see Further Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace) COMPLEX array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(1) H(2) . . . H(k), where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
* and tau in TAU(i).
*
* =====================================================================
*
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cgeqrf
USAGE:
tau, work, info, a = NumRu::Lapack.cgeqrf( m, a, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
* Purpose
* =======
*
* CGEQRF computes a QR factorization of a complex M-by-N matrix A:
* A = Q * R.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, the elements on and above the diagonal of the array
* contain the min(M,N)-by-N upper trapezoidal matrix R (R is
* upper triangular if m >= n); the elements below the diagonal,
* with the array TAU, represent the unitary matrix Q as a
* product of min(m,n) elementary reflectors (see Further
* Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N).
* For optimum performance LWORK >= N*NB, where NB is
* the optimal blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(1) H(2) . . . H(k), where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
* and tau in TAU(i).
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL CGEQR2, CLARFB, CLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
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cgeqrfp
USAGE:
tau, work, info, a = NumRu::Lapack.cgeqrfp( m, a, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGEQRFP( M, N, A, LDA, TAU, WORK, LWORK, INFO )
* Purpose
* =======
*
* CGEQRFP computes a QR factorization of a complex M-by-N matrix A:
* A = Q * R.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, the elements on and above the diagonal of the array
* contain the min(M,N)-by-N upper trapezoidal matrix R (R is
* upper triangular if m >= n); the elements below the diagonal,
* with the array TAU, represent the unitary matrix Q as a
* product of min(m,n) elementary reflectors (see Further
* Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N).
* For optimum performance LWORK >= N*NB, where NB is
* the optimal blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(1) H(2) . . . H(k), where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
* and tau in TAU(i).
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL CGEQR2P, CLARFB, CLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
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cgerfs
USAGE:
ferr, berr, info, x = NumRu::Lapack.cgerfs( trans, a, af, ipiv, b, x, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
* Purpose
* =======
*
* CGERFS improves the computed solution to a system of linear
* equations and provides error bounds and backward error estimates for
* the solution.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': A * X = B (No transpose)
* = 'T': A**T * X = B (Transpose)
* = 'C': A**H * X = B (Conjugate transpose)
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* A (input) COMPLEX array, dimension (LDA,N)
* The original N-by-N matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* AF (input) COMPLEX array, dimension (LDAF,N)
* The factors L and U from the factorization A = P*L*U
* as computed by CGETRF.
*
* LDAF (input) INTEGER
* The leading dimension of the array AF. LDAF >= max(1,N).
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices from CGETRF; for 1<=i<=N, row i of the
* matrix was interchanged with row IPIV(i).
*
* B (input) COMPLEX array, dimension (LDB,NRHS)
* The right hand side matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (input/output) COMPLEX array, dimension (LDX,NRHS)
* On entry, the solution matrix X, as computed by CGETRS.
* On exit, the improved solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* FERR (output) REAL array, dimension (NRHS)
* The estimated forward error bound for each solution vector
* X(j) (the j-th column of the solution matrix X).
* If XTRUE is the true solution corresponding to X(j), FERR(j)
* is an estimated upper bound for the magnitude of the largest
* element in (X(j) - XTRUE) divided by the magnitude of the
* largest element in X(j). The estimate is as reliable as
* the estimate for RCOND, and is almost always a slight
* overestimate of the true error.
*
* BERR (output) REAL array, dimension (NRHS)
* The componentwise relative backward error of each solution
* vector X(j) (i.e., the smallest relative change in
* any element of A or B that makes X(j) an exact solution).
*
* WORK (workspace) COMPLEX array, dimension (2*N)
*
* RWORK (workspace) REAL array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Internal Parameters
* ===================
*
* ITMAX is the maximum number of steps of iterative refinement.
*
* =====================================================================
*
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cgerfsx
USAGE:
rcond, berr, err_bnds_norm, err_bnds_comp, info, x, params = NumRu::Lapack.cgerfsx( trans, equed, a, af, ipiv, r, c, b, x, params, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
* Purpose
* =======
*
* CGERFSX improves the computed solution to a system of linear
* equations and provides error bounds and backward error estimates
* for the solution. In addition to normwise error bound, the code
* provides maximum componentwise error bound if possible. See
* comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
* error bounds.
*
* The original system of linear equations may have been equilibrated
* before calling this routine, as described by arguments EQUED, R
* and C below. In this case, the solution and error bounds returned
* are for the original unequilibrated system.
*
* Arguments
* =========
*
* Some optional parameters are bundled in the PARAMS array. These
* settings determine how refinement is performed, but often the
* defaults are acceptable. If the defaults are acceptable, users
* can pass NPARAMS = 0 which prevents the source code from accessing
* the PARAMS argument.
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': A * X = B (No transpose)
* = 'T': A**T * X = B (Transpose)
* = 'C': A**H * X = B (Conjugate transpose = Transpose)
*
* EQUED (input) CHARACTER*1
* Specifies the form of equilibration that was done to A
* before calling this routine. This is needed to compute
* the solution and error bounds correctly.
* = 'N': No equilibration
* = 'R': Row equilibration, i.e., A has been premultiplied by
* diag(R).
* = 'C': Column equilibration, i.e., A has been postmultiplied
* by diag(C).
* = 'B': Both row and column equilibration, i.e., A has been
* replaced by diag(R) * A * diag(C).
* The right hand side B has been changed accordingly.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* A (input) COMPLEX array, dimension (LDA,N)
* The original N-by-N matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* AF (input) COMPLEX array, dimension (LDAF,N)
* The factors L and U from the factorization A = P*L*U
* as computed by CGETRF.
*
* LDAF (input) INTEGER
* The leading dimension of the array AF. LDAF >= max(1,N).
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices from CGETRF; for 1<=i<=N, row i of the
* matrix was interchanged with row IPIV(i).
*
* R (input) REAL array, dimension (N)
* The row scale factors for A. If EQUED = 'R' or 'B', A is
* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
* is not accessed.
* If R is accessed, each element of R should be a power of the radix
* to ensure a reliable solution and error estimates. Scaling by
* powers of the radix does not cause rounding errors unless the
* result underflows or overflows. Rounding errors during scaling
* lead to refining with a matrix that is not equivalent to the
* input matrix, producing error estimates that may not be
* reliable.
*
* C (input) REAL array, dimension (N)
* The column scale factors for A. If EQUED = 'C' or 'B', A is
* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
* is not accessed.
* If C is accessed, each element of C should be a power of the radix
* to ensure a reliable solution and error estimates. Scaling by
* powers of the radix does not cause rounding errors unless the
* result underflows or overflows. Rounding errors during scaling
* lead to refining with a matrix that is not equivalent to the
* input matrix, producing error estimates that may not be
* reliable.
*
* B (input) COMPLEX array, dimension (LDB,NRHS)
* The right hand side matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (input/output) COMPLEX array, dimension (LDX,NRHS)
* On entry, the solution matrix X, as computed by CGETRS.
* On exit, the improved solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* RCOND (output) REAL
* Reciprocal scaled condition number. This is an estimate of the
* reciprocal Skeel condition number of the matrix A after
* equilibration (if done). If this is less than the machine
* precision (in particular, if it is zero), the matrix is singular
* to working precision. Note that the error may still be small even
* if this number is very small and the matrix appears ill-
* conditioned.
*
* BERR (output) REAL array, dimension (NRHS)
* Componentwise relative backward error. This is the
* componentwise relative backward error of each solution vector X(j)
* (i.e., the smallest relative change in any element of A or B that
* makes X(j) an exact solution).
*
* N_ERR_BNDS (input) INTEGER
* Number of error bounds to return for each right hand side
* and each type (normwise or componentwise). See ERR_BNDS_NORM and
* ERR_BNDS_COMP below.
*
* ERR_BNDS_NORM (output) REAL array, dimension (NRHS, N_ERR_BNDS)
* For each right-hand side, this array contains information about
* various error bounds and condition numbers corresponding to the
* normwise relative error, which is defined as follows:
*
* Normwise relative error in the ith solution vector:
* max_j (abs(XTRUE(j,i) - X(j,i)))
* ------------------------------
* max_j abs(X(j,i))
*
* The array is indexed by the type of error information as described
* below. There currently are up to three pieces of information
* returned.
*
* The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
* right-hand side.
*
* The second index in ERR_BNDS_NORM(:,err) contains the following
* three fields:
* err = 1 "Trust/don't trust" boolean. Trust the answer if the
* reciprocal condition number is less than the threshold
* sqrt(n) * slamch('Epsilon').
*
* err = 2 "Guaranteed" error bound: The estimated forward error,
* almost certainly within a factor of 10 of the true error
* so long as the next entry is greater than the threshold
* sqrt(n) * slamch('Epsilon'). This error bound should only
* be trusted if the previous boolean is true.
*
* err = 3 Reciprocal condition number: Estimated normwise
* reciprocal condition number. Compared with the threshold
* sqrt(n) * slamch('Epsilon') to determine if the error
* estimate is "guaranteed". These reciprocal condition
* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
* appropriately scaled matrix Z.
* Let Z = S*A, where S scales each row by a power of the
* radix so all absolute row sums of Z are approximately 1.
*
* See Lapack Working Note 165 for further details and extra
* cautions.
*
* ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS)
* For each right-hand side, this array contains information about
* various error bounds and condition numbers corresponding to the
* componentwise relative error, which is defined as follows:
*
* Componentwise relative error in the ith solution vector:
* abs(XTRUE(j,i) - X(j,i))
* max_j ----------------------
* abs(X(j,i))
*
* The array is indexed by the right-hand side i (on which the
* componentwise relative error depends), and the type of error
* information as described below. There currently are up to three
* pieces of information returned for each right-hand side. If
* componentwise accuracy is not requested (PARAMS(3) = 0.0), then
* ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
* the first (:,N_ERR_BNDS) entries are returned.
*
* The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
* right-hand side.
*
* The second index in ERR_BNDS_COMP(:,err) contains the following
* three fields:
* err = 1 "Trust/don't trust" boolean. Trust the answer if the
* reciprocal condition number is less than the threshold
* sqrt(n) * slamch('Epsilon').
*
* err = 2 "Guaranteed" error bound: The estimated forward error,
* almost certainly within a factor of 10 of the true error
* so long as the next entry is greater than the threshold
* sqrt(n) * slamch('Epsilon'). This error bound should only
* be trusted if the previous boolean is true.
*
* err = 3 Reciprocal condition number: Estimated componentwise
* reciprocal condition number. Compared with the threshold
* sqrt(n) * slamch('Epsilon') to determine if the error
* estimate is "guaranteed". These reciprocal condition
* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
* appropriately scaled matrix Z.
* Let Z = S*(A*diag(x)), where x is the solution for the
* current right-hand side and S scales each row of
* A*diag(x) by a power of the radix so all absolute row
* sums of Z are approximately 1.
*
* See Lapack Working Note 165 for further details and extra
* cautions.
*
* NPARAMS (input) INTEGER
* Specifies the number of parameters set in PARAMS. If .LE. 0, the
* PARAMS array is never referenced and default values are used.
*
* PARAMS (input / output) REAL array, dimension NPARAMS
* Specifies algorithm parameters. If an entry is .LT. 0.0, then
* that entry will be filled with default value used for that
* parameter. Only positions up to NPARAMS are accessed; defaults
* are used for higher-numbered parameters.
*
* PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
* refinement or not.
* Default: 1.0
* = 0.0 : No refinement is performed, and no error bounds are
* computed.
* = 1.0 : Use the double-precision refinement algorithm,
* possibly with doubled-single computations if the
* compilation environment does not support DOUBLE
* PRECISION.
* (other values are reserved for future use)
*
* PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
* computations allowed for refinement.
* Default: 10
* Aggressive: Set to 100 to permit convergence using approximate
* factorizations or factorizations other than LU. If
* the factorization uses a technique other than
* Gaussian elimination, the guarantees in
* err_bnds_norm and err_bnds_comp may no longer be
* trustworthy.
*
* PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
* will attempt to find a solution with small componentwise
* relative error in the double-precision algorithm. Positive
* is true, 0.0 is false.
* Default: 1.0 (attempt componentwise convergence)
*
* WORK (workspace) COMPLEX array, dimension (2*N)
*
* RWORK (workspace) REAL array, dimension (2*N)
*
* INFO (output) INTEGER
* = 0: Successful exit. The solution to every right-hand side is
* guaranteed.
* < 0: If INFO = -i, the i-th argument had an illegal value
* > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
* has been completed, but the factor U is exactly singular, so
* the solution and error bounds could not be computed. RCOND = 0
* is returned.
* = N+J: The solution corresponding to the Jth right-hand side is
* not guaranteed. The solutions corresponding to other right-
* hand sides K with K > J may not be guaranteed as well, but
* only the first such right-hand side is reported. If a small
* componentwise error is not requested (PARAMS(3) = 0.0) then
* the Jth right-hand side is the first with a normwise error
* bound that is not guaranteed (the smallest J such
* that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
* the Jth right-hand side is the first with either a normwise or
* componentwise error bound that is not guaranteed (the smallest
* J such that either ERR_BNDS_NORM(J,1) = 0.0 or
* ERR_BNDS_COMP(J,1) = 0.0). See the definition of
* ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
* about all of the right-hand sides check ERR_BNDS_NORM or
* ERR_BNDS_COMP.
*
* ==================================================================
*
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cgerq2
USAGE:
tau, info, a = NumRu::Lapack.cgerq2( a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGERQ2( M, N, A, LDA, TAU, WORK, INFO )
* Purpose
* =======
*
* CGERQ2 computes an RQ factorization of a complex m by n matrix A:
* A = R * Q.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the m by n matrix A.
* On exit, if m <= n, the upper triangle of the subarray
* A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
* if m >= n, the elements on and above the (m-n)-th subdiagonal
* contain the m by n upper trapezoidal matrix R; the remaining
* elements, with the array TAU, represent the unitary matrix
* Q as a product of elementary reflectors (see Further
* Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace) COMPLEX array, dimension (M)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(1)' H(2)' . . . H(k)', where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on
* exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i).
*
* =====================================================================
*
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cgerqf
USAGE:
tau, work, info, a = NumRu::Lapack.cgerqf( m, a, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
* Purpose
* =======
*
* CGERQF computes an RQ factorization of a complex M-by-N matrix A:
* A = R * Q.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit,
* if m <= n, the upper triangle of the subarray
* A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
* if m >= n, the elements on and above the (m-n)-th subdiagonal
* contain the M-by-N upper trapezoidal matrix R;
* the remaining elements, with the array TAU, represent the
* unitary matrix Q as a product of min(m,n) elementary
* reflectors (see Further Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,M).
* For optimum performance LWORK >= M*NB, where NB is
* the optimal blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(1)' H(2)' . . . H(k)', where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on
* exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i).
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
$ MU, NB, NBMIN, NU, NX
* ..
* .. External Subroutines ..
EXTERNAL CGERQ2, CLARFB, CLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
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cgesc2
USAGE:
scale, rhs = NumRu::Lapack.cgesc2( a, rhs, ipiv, jpiv, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGESC2( N, A, LDA, RHS, IPIV, JPIV, SCALE )
* Purpose
* =======
*
* CGESC2 solves a system of linear equations
*
* A * X = scale* RHS
*
* with a general N-by-N matrix A using the LU factorization with
* complete pivoting computed by CGETC2.
*
*
* Arguments
* =========
*
* N (input) INTEGER
* The number of columns of the matrix A.
*
* A (input) COMPLEX array, dimension (LDA, N)
* On entry, the LU part of the factorization of the n-by-n
* matrix A computed by CGETC2: A = P * L * U * Q
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1, N).
*
* RHS (input/output) COMPLEX array, dimension N.
* On entry, the right hand side vector b.
* On exit, the solution vector X.
*
* IPIV (input) INTEGER array, dimension (N).
* The pivot indices; for 1 <= i <= N, row i of the
* matrix has been interchanged with row IPIV(i).
*
* JPIV (input) INTEGER array, dimension (N).
* The pivot indices; for 1 <= j <= N, column j of the
* matrix has been interchanged with column JPIV(j).
*
* SCALE (output) REAL
* On exit, SCALE contains the scale factor. SCALE is chosen
* 0 <= SCALE <= 1 to prevent owerflow in the solution.
*
* Further Details
* ===============
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* =====================================================================
*
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cgesdd
USAGE:
s, u, vt, work, info, a = NumRu::Lapack.cgesdd( jobz, a, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, RWORK, IWORK, INFO )
* Purpose
* =======
*
* CGESDD computes the singular value decomposition (SVD) of a complex
* M-by-N matrix A, optionally computing the left and/or right singular
* vectors, by using divide-and-conquer method. The SVD is written
*
* A = U * SIGMA * conjugate-transpose(V)
*
* where SIGMA is an M-by-N matrix which is zero except for its
* min(m,n) diagonal elements, U is an M-by-M unitary matrix, and
* V is an N-by-N unitary matrix. The diagonal elements of SIGMA
* are the singular values of A; they are real and non-negative, and
* are returned in descending order. The first min(m,n) columns of
* U and V are the left and right singular vectors of A.
*
* Note that the routine returns VT = V**H, not V.
*
* The divide and conquer algorithm makes very mild assumptions about
* floating point arithmetic. It will work on machines with a guard
* digit in add/subtract, or on those binary machines without guard
* digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
* Cray-2. It could conceivably fail on hexadecimal or decimal machines
* without guard digits, but we know of none.
*
* Arguments
* =========
*
* JOBZ (input) CHARACTER*1
* Specifies options for computing all or part of the matrix U:
* = 'A': all M columns of U and all N rows of V**H are
* returned in the arrays U and VT;
* = 'S': the first min(M,N) columns of U and the first
* min(M,N) rows of V**H are returned in the arrays U
* and VT;
* = 'O': If M >= N, the first N columns of U are overwritten
* in the array A and all rows of V**H are returned in
* the array VT;
* otherwise, all columns of U are returned in the
* array U and the first M rows of V**H are overwritten
* in the array A;
* = 'N': no columns of U or rows of V**H are computed.
*
* M (input) INTEGER
* The number of rows of the input matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the input matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit,
* if JOBZ = 'O', A is overwritten with the first N columns
* of U (the left singular vectors, stored
* columnwise) if M >= N;
* A is overwritten with the first M rows
* of V**H (the right singular vectors, stored
* rowwise) otherwise.
* if JOBZ .ne. 'O', the contents of A are destroyed.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* S (output) REAL array, dimension (min(M,N))
* The singular values of A, sorted so that S(i) >= S(i+1).
*
* U (output) COMPLEX array, dimension (LDU,UCOL)
* UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
* UCOL = min(M,N) if JOBZ = 'S'.
* If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
* unitary matrix U;
* if JOBZ = 'S', U contains the first min(M,N) columns of U
* (the left singular vectors, stored columnwise);
* if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.
*
* LDU (input) INTEGER
* The leading dimension of the array U. LDU >= 1; if
* JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
*
* VT (output) COMPLEX array, dimension (LDVT,N)
* If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
* N-by-N unitary matrix V**H;
* if JOBZ = 'S', VT contains the first min(M,N) rows of
* V**H (the right singular vectors, stored rowwise);
* if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.
*
* LDVT (input) INTEGER
* The leading dimension of the array VT. LDVT >= 1; if
* JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
* if JOBZ = 'S', LDVT >= min(M,N).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= 1.
* if JOBZ = 'N', LWORK >= 2*min(M,N)+max(M,N).
* if JOBZ = 'O',
* LWORK >= 2*min(M,N)*min(M,N)+2*min(M,N)+max(M,N).
* if JOBZ = 'S' or 'A',
* LWORK >= min(M,N)*min(M,N)+2*min(M,N)+max(M,N).
* For good performance, LWORK should generally be larger.
*
* If LWORK = -1, a workspace query is assumed. The optimal
* size for the WORK array is calculated and stored in WORK(1),
* and no other work except argument checking is performed.
*
* RWORK (workspace) REAL array, dimension (MAX(1,LRWORK))
* If JOBZ = 'N', LRWORK >= 5*min(M,N).
* Otherwise,
* LRWORK >= min(M,N)*max(5*min(M,N)+7,2*max(M,N)+2*min(M,N)+1)
*
* IWORK (workspace) INTEGER array, dimension (8*min(M,N))
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: The updating process of SBDSDC did not converge.
*
* Further Details
* ===============
*
* Based on contributions by
* Ming Gu and Huan Ren, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
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cgesv
USAGE:
ipiv, info, a, b = NumRu::Lapack.cgesv( a, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGESV( N, NRHS, A, LDA, IPIV, B, LDB, INFO )
* Purpose
* =======
*
* CGESV computes the solution to a complex system of linear equations
* A * X = B,
* where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
*
* The LU decomposition with partial pivoting and row interchanges is
* used to factor A as
* A = P * L * U,
* where P is a permutation matrix, L is unit lower triangular, and U is
* upper triangular. The factored form of A is then used to solve the
* system of equations A * X = B.
*
* Arguments
* =========
*
* N (input) INTEGER
* The number of linear equations, i.e., the order of the
* matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the N-by-N coefficient matrix A.
* On exit, the factors L and U from the factorization
* A = P*L*U; the unit diagonal elements of L are not stored.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* IPIV (output) INTEGER array, dimension (N)
* The pivot indices that define the permutation matrix P;
* row i of the matrix was interchanged with row IPIV(i).
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the N-by-NRHS matrix of right hand side matrix B.
* On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, U(i,i) is exactly zero. The factorization
* has been completed, but the factor U is exactly
* singular, so the solution could not be computed.
*
* =====================================================================
*
* .. External Subroutines ..
EXTERNAL CGETRF, CGETRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
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cgesvd
USAGE:
s, u, vt, work, info, a = NumRu::Lapack.cgesvd( jobu, jobvt, a, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, RWORK, INFO )
* Purpose
* =======
*
* CGESVD computes the singular value decomposition (SVD) of a complex
* M-by-N matrix A, optionally computing the left and/or right singular
* vectors. The SVD is written
*
* A = U * SIGMA * conjugate-transpose(V)
*
* where SIGMA is an M-by-N matrix which is zero except for its
* min(m,n) diagonal elements, U is an M-by-M unitary matrix, and
* V is an N-by-N unitary matrix. The diagonal elements of SIGMA
* are the singular values of A; they are real and non-negative, and
* are returned in descending order. The first min(m,n) columns of
* U and V are the left and right singular vectors of A.
*
* Note that the routine returns V**H, not V.
*
* Arguments
* =========
*
* JOBU (input) CHARACTER*1
* Specifies options for computing all or part of the matrix U:
* = 'A': all M columns of U are returned in array U:
* = 'S': the first min(m,n) columns of U (the left singular
* vectors) are returned in the array U;
* = 'O': the first min(m,n) columns of U (the left singular
* vectors) are overwritten on the array A;
* = 'N': no columns of U (no left singular vectors) are
* computed.
*
* JOBVT (input) CHARACTER*1
* Specifies options for computing all or part of the matrix
* V**H:
* = 'A': all N rows of V**H are returned in the array VT;
* = 'S': the first min(m,n) rows of V**H (the right singular
* vectors) are returned in the array VT;
* = 'O': the first min(m,n) rows of V**H (the right singular
* vectors) are overwritten on the array A;
* = 'N': no rows of V**H (no right singular vectors) are
* computed.
*
* JOBVT and JOBU cannot both be 'O'.
*
* M (input) INTEGER
* The number of rows of the input matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the input matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit,
* if JOBU = 'O', A is overwritten with the first min(m,n)
* columns of U (the left singular vectors,
* stored columnwise);
* if JOBVT = 'O', A is overwritten with the first min(m,n)
* rows of V**H (the right singular vectors,
* stored rowwise);
* if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
* are destroyed.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* S (output) REAL array, dimension (min(M,N))
* The singular values of A, sorted so that S(i) >= S(i+1).
*
* U (output) COMPLEX array, dimension (LDU,UCOL)
* (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
* If JOBU = 'A', U contains the M-by-M unitary matrix U;
* if JOBU = 'S', U contains the first min(m,n) columns of U
* (the left singular vectors, stored columnwise);
* if JOBU = 'N' or 'O', U is not referenced.
*
* LDU (input) INTEGER
* The leading dimension of the array U. LDU >= 1; if
* JOBU = 'S' or 'A', LDU >= M.
*
* VT (output) COMPLEX array, dimension (LDVT,N)
* If JOBVT = 'A', VT contains the N-by-N unitary matrix
* V**H;
* if JOBVT = 'S', VT contains the first min(m,n) rows of
* V**H (the right singular vectors, stored rowwise);
* if JOBVT = 'N' or 'O', VT is not referenced.
*
* LDVT (input) INTEGER
* The leading dimension of the array VT. LDVT >= 1; if
* JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK.
* LWORK >= MAX(1,2*MIN(M,N)+MAX(M,N)).
* For good performance, LWORK should generally be larger.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* RWORK (workspace) REAL array, dimension (5*min(M,N))
* On exit, if INFO > 0, RWORK(1:MIN(M,N)-1) contains the
* unconverged superdiagonal elements of an upper bidiagonal
* matrix B whose diagonal is in S (not necessarily sorted).
* B satisfies A = U * B * VT, so it has the same singular
* values as A, and singular vectors related by U and VT.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if CBDSQR did not converge, INFO specifies how many
* superdiagonals of an intermediate bidiagonal form B
* did not converge to zero. See the description of RWORK
* above for details.
*
* =====================================================================
*
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cgesvx
USAGE:
x, rcond, ferr, berr, rwork, info, a, af, ipiv, equed, r, c, b = NumRu::Lapack.cgesvx( fact, trans, a, b, [:af => af, :ipiv => ipiv, :equed => equed, :r => r, :c => c, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
* Purpose
* =======
*
* CGESVX uses the LU factorization to compute the solution to a complex
* system of linear equations
* A * X = B,
* where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
*
* Error bounds on the solution and a condition estimate are also
* provided.
*
* Description
* ===========
*
* The following steps are performed:
*
* 1. If FACT = 'E', real scaling factors are computed to equilibrate
* the system:
* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
* Whether or not the system will be equilibrated depends on the
* scaling of the matrix A, but if equilibration is used, A is
* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
* or diag(C)*B (if TRANS = 'T' or 'C').
*
* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
* matrix A (after equilibration if FACT = 'E') as
* A = P * L * U,
* where P is a permutation matrix, L is a unit lower triangular
* matrix, and U is upper triangular.
*
* 3. If some U(i,i)=0, so that U is exactly singular, then the routine
* returns with INFO = i. Otherwise, the factored form of A is used
* to estimate the condition number of the matrix A. If the
* reciprocal of the condition number is less than machine precision,
* INFO = N+1 is returned as a warning, but the routine still goes on
* to solve for X and compute error bounds as described below.
*
* 4. The system of equations is solved for X using the factored form
* of A.
*
* 5. Iterative refinement is applied to improve the computed solution
* matrix and calculate error bounds and backward error estimates
* for it.
*
* 6. If equilibration was used, the matrix X is premultiplied by
* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
* that it solves the original system before equilibration.
*
* Arguments
* =========
*
* FACT (input) CHARACTER*1
* Specifies whether or not the factored form of the matrix A is
* supplied on entry, and if not, whether the matrix A should be
* equilibrated before it is factored.
* = 'F': On entry, AF and IPIV contain the factored form of A.
* If EQUED is not 'N', the matrix A has been
* equilibrated with scaling factors given by R and C.
* A, AF, and IPIV are not modified.
* = 'N': The matrix A will be copied to AF and factored.
* = 'E': The matrix A will be equilibrated if necessary, then
* copied to AF and factored.
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': A * X = B (No transpose)
* = 'T': A**T * X = B (Transpose)
* = 'C': A**H * X = B (Conjugate transpose)
*
* N (input) INTEGER
* The number of linear equations, i.e., the order of the
* matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
* not 'N', then A must have been equilibrated by the scaling
* factors in R and/or C. A is not modified if FACT = 'F' or
* 'N', or if FACT = 'E' and EQUED = 'N' on exit.
*
* On exit, if EQUED .ne. 'N', A is scaled as follows:
* EQUED = 'R': A := diag(R) * A
* EQUED = 'C': A := A * diag(C)
* EQUED = 'B': A := diag(R) * A * diag(C).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* AF (input or output) COMPLEX array, dimension (LDAF,N)
* If FACT = 'F', then AF is an input argument and on entry
* contains the factors L and U from the factorization
* A = P*L*U as computed by CGETRF. If EQUED .ne. 'N', then
* AF is the factored form of the equilibrated matrix A.
*
* If FACT = 'N', then AF is an output argument and on exit
* returns the factors L and U from the factorization A = P*L*U
* of the original matrix A.
*
* If FACT = 'E', then AF is an output argument and on exit
* returns the factors L and U from the factorization A = P*L*U
* of the equilibrated matrix A (see the description of A for
* the form of the equilibrated matrix).
*
* LDAF (input) INTEGER
* The leading dimension of the array AF. LDAF >= max(1,N).
*
* IPIV (input or output) INTEGER array, dimension (N)
* If FACT = 'F', then IPIV is an input argument and on entry
* contains the pivot indices from the factorization A = P*L*U
* as computed by CGETRF; row i of the matrix was interchanged
* with row IPIV(i).
*
* If FACT = 'N', then IPIV is an output argument and on exit
* contains the pivot indices from the factorization A = P*L*U
* of the original matrix A.
*
* If FACT = 'E', then IPIV is an output argument and on exit
* contains the pivot indices from the factorization A = P*L*U
* of the equilibrated matrix A.
*
* EQUED (input or output) CHARACTER*1
* Specifies the form of equilibration that was done.
* = 'N': No equilibration (always true if FACT = 'N').
* = 'R': Row equilibration, i.e., A has been premultiplied by
* diag(R).
* = 'C': Column equilibration, i.e., A has been postmultiplied
* by diag(C).
* = 'B': Both row and column equilibration, i.e., A has been
* replaced by diag(R) * A * diag(C).
* EQUED is an input argument if FACT = 'F'; otherwise, it is an
* output argument.
*
* R (input or output) REAL array, dimension (N)
* The row scale factors for A. If EQUED = 'R' or 'B', A is
* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
* is not accessed. R is an input argument if FACT = 'F';
* otherwise, R is an output argument. If FACT = 'F' and
* EQUED = 'R' or 'B', each element of R must be positive.
*
* C (input or output) REAL array, dimension (N)
* The column scale factors for A. If EQUED = 'C' or 'B', A is
* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
* is not accessed. C is an input argument if FACT = 'F';
* otherwise, C is an output argument. If FACT = 'F' and
* EQUED = 'C' or 'B', each element of C must be positive.
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the N-by-NRHS right hand side matrix B.
* On exit,
* if EQUED = 'N', B is not modified;
* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
* diag(R)*B;
* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
* overwritten by diag(C)*B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (output) COMPLEX array, dimension (LDX,NRHS)
* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
* to the original system of equations. Note that A and B are
* modified on exit if EQUED .ne. 'N', and the solution to the
* equilibrated system is inv(diag(C))*X if TRANS = 'N' and
* EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
* and EQUED = 'R' or 'B'.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* RCOND (output) REAL
* The estimate of the reciprocal condition number of the matrix
* A after equilibration (if done). If RCOND is less than the
* machine precision (in particular, if RCOND = 0), the matrix
* is singular to working precision. This condition is
* indicated by a return code of INFO > 0.
*
* FERR (output) REAL array, dimension (NRHS)
* The estimated forward error bound for each solution vector
* X(j) (the j-th column of the solution matrix X).
* If XTRUE is the true solution corresponding to X(j), FERR(j)
* is an estimated upper bound for the magnitude of the largest
* element in (X(j) - XTRUE) divided by the magnitude of the
* largest element in X(j). The estimate is as reliable as
* the estimate for RCOND, and is almost always a slight
* overestimate of the true error.
*
* BERR (output) REAL array, dimension (NRHS)
* The componentwise relative backward error of each solution
* vector X(j) (i.e., the smallest relative change in
* any element of A or B that makes X(j) an exact solution).
*
* WORK (workspace) COMPLEX array, dimension (2*N)
*
* RWORK (workspace/output) REAL array, dimension (2*N)
* On exit, RWORK(1) contains the reciprocal pivot growth
* factor norm(A)/norm(U). The "max absolute element" norm is
* used. If RWORK(1) is much less than 1, then the stability
* of the LU factorization of the (equilibrated) matrix A
* could be poor. This also means that the solution X, condition
* estimator RCOND, and forward error bound FERR could be
* unreliable. If factorization fails with 0 0: if INFO = i, and i is
* <= N: U(i,i) is exactly zero. The factorization has
* been completed, but the factor U is exactly
* singular, so the solution and error bounds
* could not be computed. RCOND = 0 is returned.
* = N+1: U is nonsingular, but RCOND is less than machine
* precision, meaning that the matrix is singular
* to working precision. Nevertheless, the
* solution and error bounds are computed because
* there are a number of situations where the
* computed solution can be more accurate than the
* value of RCOND would suggest.
*
* =====================================================================
*
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cgesvxx
USAGE:
x, rcond, rpvgrw, berr, err_bnds_norm, err_bnds_comp, info, a, af, ipiv, equed, r, c, b, params = NumRu::Lapack.cgesvxx( fact, trans, a, af, ipiv, equed, r, c, b, params, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
* Purpose
* =======
*
* CGESVXX uses the LU factorization to compute the solution to a
* complex system of linear equations A * X = B, where A is an
* N-by-N matrix and X and B are N-by-NRHS matrices.
*
* If requested, both normwise and maximum componentwise error bounds
* are returned. CGESVXX will return a solution with a tiny
* guaranteed error (O(eps) where eps is the working machine
* precision) unless the matrix is very ill-conditioned, in which
* case a warning is returned. Relevant condition numbers also are
* calculated and returned.
*
* CGESVXX accepts user-provided factorizations and equilibration
* factors; see the definitions of the FACT and EQUED options.
* Solving with refinement and using a factorization from a previous
* CGESVXX call will also produce a solution with either O(eps)
* errors or warnings, but we cannot make that claim for general
* user-provided factorizations and equilibration factors if they
* differ from what CGESVXX would itself produce.
*
* Description
* ===========
*
* The following steps are performed:
*
* 1. If FACT = 'E', real scaling factors are computed to equilibrate
* the system:
*
* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
*
* Whether or not the system will be equilibrated depends on the
* scaling of the matrix A, but if equilibration is used, A is
* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
* or diag(C)*B (if TRANS = 'T' or 'C').
*
* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor
* the matrix A (after equilibration if FACT = 'E') as
*
* A = P * L * U,
*
* where P is a permutation matrix, L is a unit lower triangular
* matrix, and U is upper triangular.
*
* 3. If some U(i,i)=0, so that U is exactly singular, then the
* routine returns with INFO = i. Otherwise, the factored form of A
* is used to estimate the condition number of the matrix A (see
* argument RCOND). If the reciprocal of the condition number is less
* than machine precision, the routine still goes on to solve for X
* and compute error bounds as described below.
*
* 4. The system of equations is solved for X using the factored form
* of A.
*
* 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
* the routine will use iterative refinement to try to get a small
* error and error bounds. Refinement calculates the residual to at
* least twice the working precision.
*
* 6. If equilibration was used, the matrix X is premultiplied by
* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
* that it solves the original system before equilibration.
*
* Arguments
* =========
*
* Some optional parameters are bundled in the PARAMS array. These
* settings determine how refinement is performed, but often the
* defaults are acceptable. If the defaults are acceptable, users
* can pass NPARAMS = 0 which prevents the source code from accessing
* the PARAMS argument.
*
* FACT (input) CHARACTER*1
* Specifies whether or not the factored form of the matrix A is
* supplied on entry, and if not, whether the matrix A should be
* equilibrated before it is factored.
* = 'F': On entry, AF and IPIV contain the factored form of A.
* If EQUED is not 'N', the matrix A has been
* equilibrated with scaling factors given by R and C.
* A, AF, and IPIV are not modified.
* = 'N': The matrix A will be copied to AF and factored.
* = 'E': The matrix A will be equilibrated if necessary, then
* copied to AF and factored.
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': A * X = B (No transpose)
* = 'T': A**T * X = B (Transpose)
* = 'C': A**H * X = B (Conjugate Transpose)
*
* N (input) INTEGER
* The number of linear equations, i.e., the order of the
* matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
* not 'N', then A must have been equilibrated by the scaling
* factors in R and/or C. A is not modified if FACT = 'F' or
* 'N', or if FACT = 'E' and EQUED = 'N' on exit.
*
* On exit, if EQUED .ne. 'N', A is scaled as follows:
* EQUED = 'R': A := diag(R) * A
* EQUED = 'C': A := A * diag(C)
* EQUED = 'B': A := diag(R) * A * diag(C).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* AF (input or output) COMPLEX array, dimension (LDAF,N)
* If FACT = 'F', then AF is an input argument and on entry
* contains the factors L and U from the factorization
* A = P*L*U as computed by CGETRF. If EQUED .ne. 'N', then
* AF is the factored form of the equilibrated matrix A.
*
* If FACT = 'N', then AF is an output argument and on exit
* returns the factors L and U from the factorization A = P*L*U
* of the original matrix A.
*
* If FACT = 'E', then AF is an output argument and on exit
* returns the factors L and U from the factorization A = P*L*U
* of the equilibrated matrix A (see the description of A for
* the form of the equilibrated matrix).
*
* LDAF (input) INTEGER
* The leading dimension of the array AF. LDAF >= max(1,N).
*
* IPIV (input or output) INTEGER array, dimension (N)
* If FACT = 'F', then IPIV is an input argument and on entry
* contains the pivot indices from the factorization A = P*L*U
* as computed by CGETRF; row i of the matrix was interchanged
* with row IPIV(i).
*
* If FACT = 'N', then IPIV is an output argument and on exit
* contains the pivot indices from the factorization A = P*L*U
* of the original matrix A.
*
* If FACT = 'E', then IPIV is an output argument and on exit
* contains the pivot indices from the factorization A = P*L*U
* of the equilibrated matrix A.
*
* EQUED (input or output) CHARACTER*1
* Specifies the form of equilibration that was done.
* = 'N': No equilibration (always true if FACT = 'N').
* = 'R': Row equilibration, i.e., A has been premultiplied by
* diag(R).
* = 'C': Column equilibration, i.e., A has been postmultiplied
* by diag(C).
* = 'B': Both row and column equilibration, i.e., A has been
* replaced by diag(R) * A * diag(C).
* EQUED is an input argument if FACT = 'F'; otherwise, it is an
* output argument.
*
* R (input or output) REAL array, dimension (N)
* The row scale factors for A. If EQUED = 'R' or 'B', A is
* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
* is not accessed. R is an input argument if FACT = 'F';
* otherwise, R is an output argument. If FACT = 'F' and
* EQUED = 'R' or 'B', each element of R must be positive.
* If R is output, each element of R is a power of the radix.
* If R is input, each element of R should be a power of the radix
* to ensure a reliable solution and error estimates. Scaling by
* powers of the radix does not cause rounding errors unless the
* result underflows or overflows. Rounding errors during scaling
* lead to refining with a matrix that is not equivalent to the
* input matrix, producing error estimates that may not be
* reliable.
*
* C (input or output) REAL array, dimension (N)
* The column scale factors for A. If EQUED = 'C' or 'B', A is
* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
* is not accessed. C is an input argument if FACT = 'F';
* otherwise, C is an output argument. If FACT = 'F' and
* EQUED = 'C' or 'B', each element of C must be positive.
* If C is output, each element of C is a power of the radix.
* If C is input, each element of C should be a power of the radix
* to ensure a reliable solution and error estimates. Scaling by
* powers of the radix does not cause rounding errors unless the
* result underflows or overflows. Rounding errors during scaling
* lead to refining with a matrix that is not equivalent to the
* input matrix, producing error estimates that may not be
* reliable.
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the N-by-NRHS right hand side matrix B.
* On exit,
* if EQUED = 'N', B is not modified;
* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
* diag(R)*B;
* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
* overwritten by diag(C)*B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (output) COMPLEX array, dimension (LDX,NRHS)
* If INFO = 0, the N-by-NRHS solution matrix X to the original
* system of equations. Note that A and B are modified on exit
* if EQUED .ne. 'N', and the solution to the equilibrated system is
* inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
* inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* RCOND (output) REAL
* Reciprocal scaled condition number. This is an estimate of the
* reciprocal Skeel condition number of the matrix A after
* equilibration (if done). If this is less than the machine
* precision (in particular, if it is zero), the matrix is singular
* to working precision. Note that the error may still be small even
* if this number is very small and the matrix appears ill-
* conditioned.
*
* RPVGRW (output) REAL
* Reciprocal pivot growth. On exit, this contains the reciprocal
* pivot growth factor norm(A)/norm(U). The "max absolute element"
* norm is used. If this is much less than 1, then the stability of
* the LU factorization of the (equilibrated) matrix A could be poor.
* This also means that the solution X, estimated condition numbers,
* and error bounds could be unreliable. If factorization fails with
* 0 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
* has been completed, but the factor U is exactly singular, so
* the solution and error bounds could not be computed. RCOND = 0
* is returned.
* = N+J: The solution corresponding to the Jth right-hand side is
* not guaranteed. The solutions corresponding to other right-
* hand sides K with K > J may not be guaranteed as well, but
* only the first such right-hand side is reported. If a small
* componentwise error is not requested (PARAMS(3) = 0.0) then
* the Jth right-hand side is the first with a normwise error
* bound that is not guaranteed (the smallest J such
* that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
* the Jth right-hand side is the first with either a normwise or
* componentwise error bound that is not guaranteed (the smallest
* J such that either ERR_BNDS_NORM(J,1) = 0.0 or
* ERR_BNDS_COMP(J,1) = 0.0). See the definition of
* ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
* about all of the right-hand sides check ERR_BNDS_NORM or
* ERR_BNDS_COMP.
*
* ==================================================================
*
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cgetc2
USAGE:
ipiv, jpiv, info, a = NumRu::Lapack.cgetc2( a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGETC2( N, A, LDA, IPIV, JPIV, INFO )
* Purpose
* =======
*
* CGETC2 computes an LU factorization, using complete pivoting, of the
* n-by-n matrix A. The factorization has the form A = P * L * U * Q,
* where P and Q are permutation matrices, L is lower triangular with
* unit diagonal elements and U is upper triangular.
*
* This is a level 1 BLAS version of the algorithm.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA, N)
* On entry, the n-by-n matrix to be factored.
* On exit, the factors L and U from the factorization
* A = P*L*U*Q; the unit diagonal elements of L are not stored.
* If U(k, k) appears to be less than SMIN, U(k, k) is given the
* value of SMIN, giving a nonsingular perturbed system.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1, N).
*
* IPIV (output) INTEGER array, dimension (N).
* The pivot indices; for 1 <= i <= N, row i of the
* matrix has been interchanged with row IPIV(i).
*
* JPIV (output) INTEGER array, dimension (N).
* The pivot indices; for 1 <= j <= N, column j of the
* matrix has been interchanged with column JPIV(j).
*
* INFO (output) INTEGER
* = 0: successful exit
* > 0: if INFO = k, U(k, k) is likely to produce overflow if
* one tries to solve for x in Ax = b. So U is perturbed
* to avoid the overflow.
*
* Further Details
* ===============
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* =====================================================================
*
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cgetf2
USAGE:
ipiv, info, a = NumRu::Lapack.cgetf2( m, a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGETF2( M, N, A, LDA, IPIV, INFO )
* Purpose
* =======
*
* CGETF2 computes an LU factorization of a general m-by-n matrix A
* using partial pivoting with row interchanges.
*
* The factorization has the form
* A = P * L * U
* where P is a permutation matrix, L is lower triangular with unit
* diagonal elements (lower trapezoidal if m > n), and U is upper
* triangular (upper trapezoidal if m < n).
*
* This is the right-looking Level 2 BLAS version of the algorithm.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the m by n matrix to be factored.
* On exit, the factors L and U from the factorization
* A = P*L*U; the unit diagonal elements of L are not stored.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* IPIV (output) INTEGER array, dimension (min(M,N))
* The pivot indices; for 1 <= i <= min(M,N), row i of the
* matrix was interchanged with row IPIV(i).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -k, the k-th argument had an illegal value
* > 0: if INFO = k, U(k,k) is exactly zero. The factorization
* has been completed, but the factor U is exactly
* singular, and division by zero will occur if it is used
* to solve a system of equations.
*
* =====================================================================
*
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cgetrf
USAGE:
ipiv, info, a = NumRu::Lapack.cgetrf( m, a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGETRF( M, N, A, LDA, IPIV, INFO )
* Purpose
* =======
*
* CGETRF computes an LU factorization of a general M-by-N matrix A
* using partial pivoting with row interchanges.
*
* The factorization has the form
* A = P * L * U
* where P is a permutation matrix, L is lower triangular with unit
* diagonal elements (lower trapezoidal if m > n), and U is upper
* triangular (upper trapezoidal if m < n).
*
* This is the right-looking Level 3 BLAS version of the algorithm.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix to be factored.
* On exit, the factors L and U from the factorization
* A = P*L*U; the unit diagonal elements of L are not stored.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* IPIV (output) INTEGER array, dimension (min(M,N))
* The pivot indices; for 1 <= i <= min(M,N), row i of the
* matrix was interchanged with row IPIV(i).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, U(i,i) is exactly zero. The factorization
* has been completed, but the factor U is exactly
* singular, and division by zero will occur if it is used
* to solve a system of equations.
*
* =====================================================================
*
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cgetri
USAGE:
work, info, a = NumRu::Lapack.cgetri( a, ipiv, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGETRI( N, A, LDA, IPIV, WORK, LWORK, INFO )
* Purpose
* =======
*
* CGETRI computes the inverse of a matrix using the LU factorization
* computed by CGETRF.
*
* This method inverts U and then computes inv(A) by solving the system
* inv(A)*L = inv(U) for inv(A).
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the factors L and U from the factorization
* A = P*L*U as computed by CGETRF.
* On exit, if INFO = 0, the inverse of the original matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices from CGETRF; for 1<=i<=N, row i of the
* matrix was interchanged with row IPIV(i).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO=0, then WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N).
* For optimal performance LWORK >= N*NB, where NB is
* the optimal blocksize returned by ILAENV.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, U(i,i) is exactly zero; the matrix is
* singular and its inverse could not be computed.
*
* =====================================================================
*
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cgetrs
USAGE:
info, b = NumRu::Lapack.cgetrs( trans, a, ipiv, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CGETRS( TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
* Purpose
* =======
*
* CGETRS solves a system of linear equations
* A * X = B, A**T * X = B, or A**H * X = B
* with a general N-by-N matrix A using the LU factorization computed
* by CGETRF.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': A * X = B (No transpose)
* = 'T': A**T * X = B (Transpose)
* = 'C': A**H * X = B (Conjugate transpose)
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* A (input) COMPLEX array, dimension (LDA,N)
* The factors L and U from the factorization A = P*L*U
* as computed by CGETRF.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices from CGETRF; for 1<=i<=N, row i of the
* matrix was interchanged with row IPIV(i).
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the right hand side matrix B.
* On exit, the solution matrix X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
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