COMPLEX routines for triangular, packed storage matrix
ctpcon
USAGE:
rcond, info = NumRu::Lapack.ctpcon( norm, uplo, diag, ap, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CTPCON( NORM, UPLO, DIAG, N, AP, RCOND, WORK, RWORK, INFO )
* Purpose
* =======
*
* CTPCON estimates the reciprocal of the condition number of a packed
* triangular matrix A, in either the 1-norm or the infinity-norm.
*
* The norm of A is computed and an estimate is obtained for
* norm(inv(A)), then 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.
*
* UPLO (input) CHARACTER*1
* = 'U': A is upper triangular;
* = 'L': A is lower triangular.
*
* DIAG (input) CHARACTER*1
* = 'N': A is non-unit triangular;
* = 'U': A is unit triangular.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* AP (input) COMPLEX array, dimension (N*(N+1)/2)
* The upper or lower triangular matrix A, packed columnwise in
* a linear array. The j-th column of A is stored in the array
* AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
* If DIAG = 'U', the diagonal elements of A are not referenced
* and are assumed to be 1.
*
* 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 (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
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ctprfs
USAGE:
ferr, berr, info = NumRu::Lapack.ctprfs( uplo, trans, diag, ap, b, x, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
* Purpose
* =======
*
* CTPRFS provides error bounds and backward error estimates for the
* solution to a system of linear equations with a triangular packed
* coefficient matrix.
*
* The solution matrix X must be computed by CTPTRS or some other
* means before entering this routine. CTPRFS does not do iterative
* refinement because doing so cannot improve the backward error.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': A is upper triangular;
* = 'L': A is lower triangular.
*
* 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)
*
* DIAG (input) CHARACTER*1
* = 'N': A is non-unit triangular;
* = 'U': A is unit triangular.
*
* 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.
*
* AP (input) COMPLEX array, dimension (N*(N+1)/2)
* The upper or lower triangular matrix A, packed columnwise in
* a linear array. The j-th column of A is stored in the array
* AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
* If DIAG = 'U', the diagonal elements of A are not referenced
* and are assumed to be 1.
*
* 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) COMPLEX array, dimension (LDX,NRHS)
* The 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
*
* =====================================================================
*
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ctptri
USAGE:
info, ap = NumRu::Lapack.ctptri( uplo, diag, n, ap, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CTPTRI( UPLO, DIAG, N, AP, INFO )
* Purpose
* =======
*
* CTPTRI computes the inverse of a complex upper or lower triangular
* matrix A stored in packed format.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': A is upper triangular;
* = 'L': A is lower triangular.
*
* DIAG (input) CHARACTER*1
* = 'N': A is non-unit triangular;
* = 'U': A is unit triangular.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* AP (input/output) COMPLEX array, dimension (N*(N+1)/2)
* On entry, the upper or lower triangular matrix A, stored
* columnwise in a linear array. The j-th column of A is stored
* in the array AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n.
* See below for further details.
* On exit, the (triangular) inverse of the original matrix, in
* the same packed storage format.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, A(i,i) is exactly zero. The triangular
* matrix is singular and its inverse can not be computed.
*
* Further Details
* ===============
*
* A triangular matrix A can be transferred to packed storage using one
* of the following program segments:
*
* UPLO = 'U': UPLO = 'L':
*
* JC = 1 JC = 1
* DO 2 J = 1, N DO 2 J = 1, N
* DO 1 I = 1, J DO 1 I = J, N
* AP(JC+I-1) = A(I,J) AP(JC+I-J) = A(I,J)
* 1 CONTINUE 1 CONTINUE
* JC = JC + J JC = JC + N - J + 1
* 2 CONTINUE 2 CONTINUE
*
* =====================================================================
*
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ctptrs
USAGE:
info, b = NumRu::Lapack.ctptrs( uplo, trans, diag, n, ap, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CTPTRS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, INFO )
* Purpose
* =======
*
* CTPTRS solves a triangular system of the form
*
* A * X = B, A**T * X = B, or A**H * X = B,
*
* where A is a triangular matrix of order N stored in packed format,
* and B is an N-by-NRHS matrix. A check is made to verify that A is
* nonsingular.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': A is upper triangular;
* = 'L': A is lower triangular.
*
* 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)
*
* DIAG (input) CHARACTER*1
* = 'N': A is non-unit triangular;
* = 'U': A is unit triangular.
*
* 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.
*
* AP (input) COMPLEX array, dimension (N*(N+1)/2)
* The upper or lower triangular matrix A, packed columnwise in
* a linear array. The j-th column of A is stored in the array
* AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the right hand side matrix B.
* On exit, if INFO = 0, 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
* > 0: if INFO = i, the i-th diagonal element of A is zero,
* indicating that the matrix is singular and the
* solutions X have not been computed.
*
* =====================================================================
*
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ctpttf
USAGE:
arf, info = NumRu::Lapack.ctpttf( transr, uplo, n, ap, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CTPTTF( TRANSR, UPLO, N, AP, ARF, INFO )
* Purpose
* =======
*
* CTPTTF copies a triangular matrix A from standard packed format (TP)
* to rectangular full packed format (TF).
*
* Arguments
* =========
*
* TRANSR (input) CHARACTER*1
* = 'N': ARF in Normal format is wanted;
* = 'C': ARF in Conjugate-transpose format is wanted.
*
* UPLO (input) CHARACTER*1
* = 'U': A is upper triangular;
* = 'L': A is lower triangular.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* AP (input) COMPLEX array, dimension ( N*(N+1)/2 ),
* On entry, the upper or lower triangular matrix A, packed
* columnwise in a linear array. The j-th column of A is stored
* in the array AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*
* ARF (output) COMPLEX array, dimension ( N*(N+1)/2 ),
* On exit, the upper or lower triangular matrix A stored in
* RFP format. For a further discussion see Notes below.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* We first consider Standard Packed Format when N is even.
* We give an example where N = 6.
*
* AP is Upper AP is Lower
*
* 00 01 02 03 04 05 00
* 11 12 13 14 15 10 11
* 22 23 24 25 20 21 22
* 33 34 35 30 31 32 33
* 44 45 40 41 42 43 44
* 55 50 51 52 53 54 55
*
*
* Let TRANSR = 'N'. RFP holds AP as follows:
* For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
* three columns of AP upper. The lower triangle A(4:6,0:2) consists of
* conjugate-transpose of the first three columns of AP upper.
* For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
* three columns of AP lower. The upper triangle A(0:2,0:2) consists of
* conjugate-transpose of the last three columns of AP lower.
* To denote conjugate we place -- above the element. This covers the
* case N even and TRANSR = 'N'.
*
* RFP A RFP A
*
* -- -- --
* 03 04 05 33 43 53
* -- --
* 13 14 15 00 44 54
* --
* 23 24 25 10 11 55
*
* 33 34 35 20 21 22
* --
* 00 44 45 30 31 32
* -- --
* 01 11 55 40 41 42
* -- -- --
* 02 12 22 50 51 52
*
* Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
* transpose of RFP A above. One therefore gets:
*
*
* RFP A RFP A
*
* -- -- -- -- -- -- -- -- -- --
* 03 13 23 33 00 01 02 33 00 10 20 30 40 50
* -- -- -- -- -- -- -- -- -- --
* 04 14 24 34 44 11 12 43 44 11 21 31 41 51
* -- -- -- -- -- -- -- -- -- --
* 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*
*
* We next consider Standard Packed Format when N is odd.
* We give an example where N = 5.
*
* AP is Upper AP is Lower
*
* 00 01 02 03 04 00
* 11 12 13 14 10 11
* 22 23 24 20 21 22
* 33 34 30 31 32 33
* 44 40 41 42 43 44
*
*
* Let TRANSR = 'N'. RFP holds AP as follows:
* For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
* three columns of AP upper. The lower triangle A(3:4,0:1) consists of
* conjugate-transpose of the first two columns of AP upper.
* For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
* three columns of AP lower. The upper triangle A(0:1,1:2) consists of
* conjugate-transpose of the last two columns of AP lower.
* To denote conjugate we place -- above the element. This covers the
* case N odd and TRANSR = 'N'.
*
* RFP A RFP A
*
* -- --
* 02 03 04 00 33 43
* --
* 12 13 14 10 11 44
*
* 22 23 24 20 21 22
* --
* 00 33 34 30 31 32
* -- --
* 01 11 44 40 41 42
*
* Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
* transpose of RFP A above. One therefore gets:
*
*
* RFP A RFP A
*
* -- -- -- -- -- -- -- -- --
* 02 12 22 00 01 00 10 20 30 40 50
* -- -- -- -- -- -- -- -- --
* 03 13 23 33 11 33 11 21 31 41 51
* -- -- -- -- -- -- -- -- --
* 04 14 24 34 44 43 44 22 32 42 52
*
* =====================================================================
*
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ctpttr
USAGE:
a, info = NumRu::Lapack.ctpttr( uplo, ap, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CTPTTR( UPLO, N, AP, A, LDA, INFO )
* Purpose
* =======
*
* CTPTTR copies a triangular matrix A from standard packed format (TP)
* to standard full format (TR).
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': A is upper triangular.
* = 'L': A is lower triangular.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* AP (input) COMPLEX array, dimension ( N*(N+1)/2 ),
* On entry, the upper or lower triangular matrix A, packed
* columnwise in a linear array. The j-th column of A is stored
* in the array AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*
* A (output) COMPLEX array, dimension ( LDA, N )
* On exit, the triangular matrix A. If UPLO = 'U', the leading
* N-by-N upper triangular part of A contains the upper
* triangular part of the matrix A, and the strictly lower
* triangular part of A is not referenced. If UPLO = 'L', the
* leading N-by-N lower triangular part of A contains the lower
* triangular part of the matrix A, and the strictly upper
* triangular part of A is not referenced.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= 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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