REAL routines for symmetric or Hermitian positive definite tridiagonal matrix
sptcon
USAGE:
rcond, info = NumRu::Lapack.sptcon( d, e, anorm, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPTCON( N, D, E, ANORM, RCOND, WORK, INFO )
* Purpose
* =======
*
* SPTCON computes the reciprocal of the condition number (in the
* 1-norm) of a real symmetric positive definite tridiagonal matrix
* using the factorization A = L*D*L**T or A = U**T*D*U computed by
* SPTTRF.
*
* Norm(inv(A)) is computed by a direct method, and the reciprocal of
* the condition number is computed as
* RCOND = 1 / (ANORM * norm(inv(A))).
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the diagonal matrix D from the
* factorization of A, as computed by SPTTRF.
*
* E (input) REAL array, dimension (N-1)
* The (n-1) off-diagonal elements of the unit bidiagonal factor
* U or L from the factorization of A, as computed by SPTTRF.
*
* ANORM (input) REAL
* The 1-norm of the original matrix A.
*
* RCOND (output) REAL
* The reciprocal of the condition number of the matrix A,
* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
* 1-norm of inv(A) computed in this routine.
*
* WORK (workspace) REAL array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The method used is described in Nicholas J. Higham, "Efficient
* Algorithms for Computing the Condition Number of a Tridiagonal
* Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
*
* =====================================================================
*
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spteqr
USAGE:
info, d, e, z = NumRu::Lapack.spteqr( compz, d, e, z, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
* Purpose
* =======
*
* SPTEQR computes all eigenvalues and, optionally, eigenvectors of a
* symmetric positive definite tridiagonal matrix by first factoring the
* matrix using SPTTRF, and then calling SBDSQR to compute the singular
* values of the bidiagonal factor.
*
* This routine computes the eigenvalues of the positive definite
* tridiagonal matrix to high relative accuracy. This means that if the
* eigenvalues range over many orders of magnitude in size, then the
* small eigenvalues and corresponding eigenvectors will be computed
* more accurately than, for example, with the standard QR method.
*
* The eigenvectors of a full or band symmetric positive definite matrix
* can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to
* reduce this matrix to tridiagonal form. (The reduction to tridiagonal
* form, however, may preclude the possibility of obtaining high
* relative accuracy in the small eigenvalues of the original matrix, if
* these eigenvalues range over many orders of magnitude.)
*
* Arguments
* =========
*
* COMPZ (input) CHARACTER*1
* = 'N': Compute eigenvalues only.
* = 'V': Compute eigenvectors of original symmetric
* matrix also. Array Z contains the orthogonal
* matrix used to reduce the original matrix to
* tridiagonal form.
* = 'I': Compute eigenvectors of tridiagonal matrix also.
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input/output) REAL array, dimension (N)
* On entry, the n diagonal elements of the tridiagonal
* matrix.
* On normal exit, D contains the eigenvalues, in descending
* order.
*
* E (input/output) REAL array, dimension (N-1)
* On entry, the (n-1) subdiagonal elements of the tridiagonal
* matrix.
* On exit, E has been destroyed.
*
* Z (input/output) REAL array, dimension (LDZ, N)
* On entry, if COMPZ = 'V', the orthogonal matrix used in the
* reduction to tridiagonal form.
* On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
* original symmetric matrix;
* if COMPZ = 'I', the orthonormal eigenvectors of the
* tridiagonal matrix.
* If INFO > 0 on exit, Z contains the eigenvectors associated
* with only the stored eigenvalues.
* If COMPZ = 'N', then Z is not referenced.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1, and if
* COMPZ = 'V' or 'I', LDZ >= max(1,N).
*
* WORK (workspace) REAL array, dimension (4*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 Cholesky factorization of the matrix could
* not be performed because the i-th principal minor
* was not positive definite.
* > N the SVD algorithm failed to converge;
* if INFO = N+i, i off-diagonal elements of the
* bidiagonal factor did not converge to zero.
*
* =====================================================================
*
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sptrfs
USAGE:
ferr, berr, info, x = NumRu::Lapack.sptrfs( d, e, df, ef, b, x, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO )
* Purpose
* =======
*
* SPTRFS improves the computed solution to a system of linear
* equations when the coefficient matrix is symmetric positive definite
* and tridiagonal, and provides error bounds and backward error
* estimates for the solution.
*
* Arguments
* =========
*
* 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.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the tridiagonal matrix A.
*
* E (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of the tridiagonal matrix A.
*
* DF (input) REAL array, dimension (N)
* The n diagonal elements of the diagonal matrix D from the
* factorization computed by SPTTRF.
*
* EF (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of the unit bidiagonal factor
* L from the factorization computed by SPTTRF.
*
* B (input) REAL 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) REAL array, dimension (LDX,NRHS)
* On entry, the solution matrix X, as computed by SPTTRS.
* 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 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).
*
* 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) REAL array, dimension (2*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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sptsv
USAGE:
info, d, e, b = NumRu::Lapack.sptsv( d, e, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPTSV( N, NRHS, D, E, B, LDB, INFO )
* Purpose
* =======
*
* SPTSV computes the solution to a real system of linear equations
* A*X = B, where A is an N-by-N symmetric positive definite tridiagonal
* matrix, and X and B are N-by-NRHS matrices.
*
* A is factored as A = L*D*L**T, and the factored form of A is then
* used to solve the system of equations.
*
* Arguments
* =========
*
* 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.
*
* D (input/output) REAL array, dimension (N)
* On entry, the n diagonal elements of the tridiagonal matrix
* A. On exit, the n diagonal elements of the diagonal matrix
* D from the factorization A = L*D*L**T.
*
* E (input/output) REAL array, dimension (N-1)
* On entry, the (n-1) subdiagonal elements of the tridiagonal
* matrix A. On exit, the (n-1) subdiagonal elements of the
* unit bidiagonal factor L from the L*D*L**T factorization of
* A. (E can also be regarded as the superdiagonal of the unit
* bidiagonal factor U from the U**T*D*U factorization of A.)
*
* B (input/output) REAL array, dimension (LDB,NRHS)
* On entry, the N-by-NRHS 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, the leading minor of order i is not
* positive definite, and the solution has not been
* computed. The factorization has not been completed
* unless i = N.
*
* =====================================================================
*
* .. External Subroutines ..
EXTERNAL SPTTRF, SPTTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
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sptsvx
USAGE:
x, rcond, ferr, berr, info, df, ef = NumRu::Lapack.sptsvx( fact, d, e, df, ef, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, INFO )
* Purpose
* =======
*
* SPTSVX uses the factorization A = L*D*L**T to compute the solution
* to a real system of linear equations A*X = B, where A is an N-by-N
* symmetric positive definite tridiagonal 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 = 'N', the matrix A is factored as A = L*D*L**T, where L
* is a unit lower bidiagonal matrix and D is diagonal. The
* factorization can also be regarded as having the form
* A = U**T*D*U.
*
* 2. If the leading i-by-i principal minor is not positive definite,
* 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.
*
* 3. The system of equations is solved for X using the factored form
* of A.
*
* 4. Iterative refinement is applied to improve the computed solution
* matrix and calculate error bounds and backward error estimates
* for it.
*
* Arguments
* =========
*
* FACT (input) CHARACTER*1
* Specifies whether or not the factored form of A has been
* supplied on entry.
* = 'F': On entry, DF and EF contain the factored form of A.
* D, E, DF, and EF will not be modified.
* = 'N': The matrix A will be copied to DF and EF and
* factored.
*
* 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.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the tridiagonal matrix A.
*
* E (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of the tridiagonal matrix A.
*
* DF (input or output) REAL array, dimension (N)
* If FACT = 'F', then DF is an input argument and on entry
* contains the n diagonal elements of the diagonal matrix D
* from the L*D*L**T factorization of A.
* If FACT = 'N', then DF is an output argument and on exit
* contains the n diagonal elements of the diagonal matrix D
* from the L*D*L**T factorization of A.
*
* EF (input or output) REAL array, dimension (N-1)
* If FACT = 'F', then EF is an input argument and on entry
* contains the (n-1) subdiagonal elements of the unit
* bidiagonal factor L from the L*D*L**T factorization of A.
* If FACT = 'N', then EF is an output argument and on exit
* contains the (n-1) subdiagonal elements of the unit
* bidiagonal factor L from the L*D*L**T factorization of A.
*
* B (input) REAL array, dimension (LDB,NRHS)
* The N-by-NRHS right hand side matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (output) REAL array, dimension (LDX,NRHS)
* If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* RCOND (output) REAL
* The reciprocal condition number of the matrix A. 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 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).
*
* 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) 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, and i is
* <= N: the leading minor of order i of A is
* not positive definite, so the factorization
* could not be completed, and the solution has not
* been 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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spttrf
USAGE:
info, d, e = NumRu::Lapack.spttrf( d, e, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPTTRF( N, D, E, INFO )
* Purpose
* =======
*
* SPTTRF computes the L*D*L' factorization of a real symmetric
* positive definite tridiagonal matrix A. The factorization may also
* be regarded as having the form A = U'*D*U.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* D (input/output) REAL array, dimension (N)
* On entry, the n diagonal elements of the tridiagonal matrix
* A. On exit, the n diagonal elements of the diagonal matrix
* D from the L*D*L' factorization of A.
*
* E (input/output) REAL array, dimension (N-1)
* On entry, the (n-1) subdiagonal elements of the tridiagonal
* matrix A. On exit, the (n-1) subdiagonal elements of the
* unit bidiagonal factor L from the L*D*L' factorization of A.
* E can also be regarded as the superdiagonal of the unit
* bidiagonal factor U from the U'*D*U factorization of A.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -k, the k-th argument had an illegal value
* > 0: if INFO = k, the leading minor of order k is not
* positive definite; if k < N, the factorization could not
* be completed, while if k = N, the factorization was
* completed, but D(N) <= 0.
*
* =====================================================================
*
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spttrs
USAGE:
info, b = NumRu::Lapack.spttrs( d, e, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPTTRS( N, NRHS, D, E, B, LDB, INFO )
* Purpose
* =======
*
* SPTTRS solves a tridiagonal system of the form
* A * X = B
* using the L*D*L' factorization of A computed by SPTTRF. D is a
* diagonal matrix specified in the vector D, L is a unit bidiagonal
* matrix whose subdiagonal is specified in the vector E, and X and B
* are N by NRHS matrices.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the tridiagonal 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.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the diagonal matrix D from the
* L*D*L' factorization of A.
*
* E (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of the unit bidiagonal factor
* L from the L*D*L' factorization of A. E can also be regarded
* as the superdiagonal of the unit bidiagonal factor U from the
* factorization A = U'*D*U.
*
* B (input/output) REAL array, dimension (LDB,NRHS)
* On entry, the right hand side vectors B for the system of
* linear equations.
* On exit, the solution vectors, X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -k, the k-th argument had an illegal value
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER J, JB, NB
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. External Subroutines ..
EXTERNAL SPTTS2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
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sptts2
USAGE:
b = NumRu::Lapack.sptts2( d, e, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPTTS2( N, NRHS, D, E, B, LDB )
* Purpose
* =======
*
* SPTTS2 solves a tridiagonal system of the form
* A * X = B
* using the L*D*L' factorization of A computed by SPTTRF. D is a
* diagonal matrix specified in the vector D, L is a unit bidiagonal
* matrix whose subdiagonal is specified in the vector E, and X and B
* are N by NRHS matrices.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the tridiagonal 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.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the diagonal matrix D from the
* L*D*L' factorization of A.
*
* E (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of the unit bidiagonal factor
* L from the L*D*L' factorization of A. E can also be regarded
* as the superdiagonal of the unit bidiagonal factor U from the
* factorization A = U'*D*U.
*
* B (input/output) REAL array, dimension (LDB,NRHS)
* On entry, the right hand side vectors B for the system of
* linear equations.
* On exit, the solution vectors, X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. External Subroutines ..
EXTERNAL SSCAL
* ..
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