sglaux - Man Page

subroutine sla_itranspose (k, kh, l, lh, x, ldx)
Solver for 4-by-4 Linear Systems.
subroutine sla_qtrmm2 (side, trans, m, n, alpha, a, lda, b, ldb, c, ldc)
Multiply with the sub-diagonal entries of a quasi triangular matrix.
subroutine sla_small_solve4 (n, a, rhs, info, eps, smlnum)
Solver for 4-by-4 Linear Systems.
subroutine sla_small_solve8 (n, a, rhs, info, eps, smlnum)
Solver for 8-by-8 Linear Systems.
subroutine sla_sort_ev (n, a, lda, q, ldq, nb, work, info)
Block Reordering of generalized Eigenvalues.
subroutine sla_sort_gev (n, a, lda, b, ldb, q, ldq, z, ldz, nb, work, lwork, info)
Block Reordering of generalized Eigenvalues.
subroutine sla_transform_general (trans, m, n, x, ldx, qa, ldqa, qb, ldqb, work)
Transform the Right-Hand Side or Solution for generalized projections.
subroutine sla_transform_standard (trans, m, x, ldx, q, ldq, work)
Transform the Right-Hand Side or Solution for standard projections.

Solver for 4-by-4 Linear Systems.

Purpose:

 SLA_ITRANSPOSE performs the implicit transpose

    X(L:LH,K:KH ) <- X(K:KH,L:LH)**T                                                   (1)

Parameters

K

          K is INTEGER
          First index of the transpose operations as in (1)

KH

          KH is INTEGER
          Second index of the transpose operations as in (1)

L

          L is INTEGER
          Third index of the transpose operations as in (1)

LH

          LH is INTEGER
          Fourth index of the transpose operations as in (1)

X

          X is REAL array, dimension (LDX,*)
          X is the matrix to perform the transpose on.

LDX

             LDX  is  INTEGER
             The leading dimension of the array X.  LDX >= MAX(1,KH,LH).

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 86 of file sla_itranspose.f90.

Multiply with the sub-diagonal entries of a quasi triangular matrix.

Purpose:

     Multiply a matrix with the sub-diagonal entries of a quasi-triangular matrix. 
     Consider a quasi-triangular matrix A where sub(A) is the matrix only containing 
     the sub-diagonal entries. Than the SLA_QTRMM2 routine computes 
 
        C = C + ALPHA * op(sub(A)) * B                                                (1) 
 
     or 
 
        C = C + ALPHA * B * op(sub(A))                                                (2)

Attention

The routine does not check the input arguments.

Parameters

SIDE

          SIDE is CHARACTER(1)
           On entry,  SIDE specifies whether  op( A ) multiplies B from
           the left or right as follows:

              SIDE = 'L' or 'l'   B := alpha*op( A )*B.

              SIDE = 'R' or 'r'   B := alpha*B*op( A ).

TRANS

          TRANS is CHARACTER(1)
           On entry, TRANS specifies the form of op( A ) to be used in
           the matrix multiplication as follows:

              TRANS == 'N' or 'n'   op( A ) = A.

              TRANS == 'T' or 't'   op( A ) = A**T.

              TRANS = 'C' or 'c'   op( A ) = A**T.

M

          M is INTEGER
           On entry, M specifies the number of rows of B. M must be at
           least zero.

N

          N is INTEGER
           On entry, N specifies the number of columns of B.  N must be
           at least zero.

ALPHA

          ALPHA is REAL.
           On entry,  ALPHA specifies the scalar  alpha. When  alpha is
           zero then  A is not referenced and  B need not be set before
           entry.

A

           A is REAL array of DIMENSION ( LDA, k ), where k is m
           when  SIDE = 'L' or 'l'  and is  n  when  SIDE = 'R' or 'r'.
           The  leading  k by k 
           upper triangular part of the array  A must contain the upper
           quasi triangular matrix.

LDA

          LDA is INTEGER
           On entry, LDA specifies the first dimension of A as declared
           in the calling (sub) program.  When  SIDE = 'L' or 'l'  then
           LDA  must be at least  max( 1, m ),  when  SIDE = 'R' or 'r'
           then LDA must be at least max( 1, n ).

B

           B is REAL array of DIMENSION ( LDB, n ).
           Before entry,  the leading  m by n part of the array  B must
           contain the matrix  B.

LDB

          LDB is INTEGER
           On entry, LDB specifies the first dimension of B as declared
           in  the  calling  (sub)  program.   LDB  must  be  at  least
           max( 1, m ).

C

           C is REAL array of DIMENSION ( LDC, n ).
           Before entry,  the leading  m by n part of the array  C must
           contain the matrix  C,  and  on exit  is overwritten  by the
           update January 2021

LDC

          LDC is INTEGER
           On entry, LDC specifies the first dimension of C as declared
           in  the  calling  (sub)  program.   LDB  must  be  at  least
           max( 1, m ).

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 153 of file sla_qtrmm2.f90.

Solver for 4-by-4 Linear Systems.

Purpose:

 SLA_SMALL_SOLVE4 solves a linear system

    A * x = b                                                                          (1)

 where A is a most a 4 by 4 matrix.

Parameters

N

          N is INTEGER
          The order of the matrix A  N = (1,2,4).

A

          A is REAL array, dimension (4,4)
          The matrix A must be stored in a 4-by-4 matrix even if it is smaller.
          On output the matrix  A is destroyed.

RHS

          RHS is REAL array, dimension (4)
          The matrix RHS contains the right hand side on input and the solution X on output if INFO == 0

INFO

          INFO is INTEGER
          == 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  The equation is not solved correctly. One of the arising inner
                system got singular.

EPS

          EPS is REAL
          EPS is the machine epsilon determined by DLAMCH

SMLNUM

          SMLNUM is REAL
          SMLNUM is the smallest number in REAL which can be represented without underflow.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 98 of file sla_small_solve4.f90.

Solver for 8-by-8 Linear Systems.

Purpose:

 SLA_SMALL_SOLVE8 solves a linear system

    A * x = b                                                                          (1)

 where A is a most a 8 by 8 matrix with leading dimension 8.

Parameters

N

          N is INTEGER
          The order of the matrix A  N = {1,2,4,8} .

A

          A is REAL array, dimension (8,8)
          The matrix A must be stored in a 4-by-4 matrix even if it is smaller.
          On output the matrix  A is destroyed.

RHS

          RHS is REAL array, dimension (8)
          The matrix RHS contains the right hand side on input and the solution X on output if INFO == 0

INFO

          INFO is INTEGER
          == 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  The equation is not solved correctly. One of the arising inner
                system got singular.

EPS

          EPS is REAL
          EPS is the machine epsilon determined by DLAMCH

SMLNUM

          SMLNUM is REAL
          SMLNUM is the smallest number in REAL which can be represented without underflow.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 98 of file sla_small_solve8.f90.

Block Reordering of generalized Eigenvalues.

Purpose:

     Reorder the eigenvalues of a matrix A such that
     the complex eigenvalues pairs will not cause a change of the block
     size in blocked linear matrix equation solver codes.

     The function uses STREXC from LAPACK for moving the eigenvalues
     on the diagonal.

Parameters

N

          N is INTEGER
          The order of the matrices A and Q, N >= 0.

A

             A  is a REAL array, dimension (LDA,N)
             On entry, the leading N-by-N upper
             Hessenberg part of this array must contain the
             generalized Schur factor A_s of the matrix A.
             On exit, the leading N-by-N part of this array contains
             the generalized Schur factor A_s of the matrix A with
             no complex eigenvalues pairs an A(k*NB,k*NB), where k is
             a non negative integer.

LDA

             LDA is INTEGER
             The leading dimension of the array A.  LDA >= MAX(1,N).

Q

             Q is a REAL array, dimension (LDQ,N)
             On entry, the leading N-by-N part of
             this array must contain the orthogonal matrix Q from
             the generalized Schur factorization.
             On exit, the leading N-by-N part of this array contains
             the update January 2021
             factorization with integrated eigenvalue reordering.

LDQ

             LDQ  is  INTEGER
             The leading dimension of the array Q.  LDQ >= MAX(1,N).

NB

            NB is INTEGER
             Block size of the solver planed to use. Typical values
             are 16, 32, or 64. The block size must be an even positive
             integer otherwise an error is returned.

WORK

             WORK is INTEGER array, dimension (N)
             Workspace

INFO

             INFO is INTEGER
             == 0:  successful exit;
             < 0:  if INFO = -i, the i-th argument had an illegal
                   value;
             = 1:  The internal STGEX2 caused an error. ]

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 124 of file sla_sort_ev.f90.

Block Reordering of generalized Eigenvalues.

Purpose:

     Reorder the eigenvalues of a matrix pencil (A,B) such that
     the complex eigenvalues pairs will not cause a change of the block
     size in blocked linear matrix equation solver codes.

     The function uses STGEX2 from LAPACK for moving the eigenvalues
     on the diagonal.

Parameters

N

          N is INTEGER
          The order of the matrices A, B, Q, and Z, N >= 0.

A

             A  is a REAL array, dimension (LDA,N)
             On entry, the leading N-by-N upper
             Hessenberg part of this array must contain the
             generalized Schur factor A_s of the matrix A.
             On exit, the leading N-by-N part of this array contains
             the generalized Schur factor A_s of the matrix A with
             no complex eigenvalues pairs an A(k*NB,k*NB), where k is
             a non negative integer.

LDA

             LDA is INTEGER
             The leading dimension of the array A.  LDA >= MAX(1,N).

B

             B is a REAL array, dimension (LDB,N)
             On entry, the leading N-by-N upper
             triangular part of this array must contain the
             generalized Schur factor B_s of the matrix B,
             On exit, the leading N-by-N part of this array contains
             the generalized Schur factor B_s of the matrix B
             with reordered eigenvalues.

LDB

             LDB is INTEGER
             The leading dimension of the array B.  LDB >= MAX(1,N).

Q

             Q is a REAL array, dimension (LDQ,N)
             On entry, the leading N-by-N part of
             this array must contain the orthogonal matrix Q from
             the generalized Schur factorization.
             On exit, the leading N-by-N part of this array contains
             the update January 2021
             factorization with integrated eigenvalue reordering.

LDQ

             LDQ  is  INTEGER
             The leading dimension of the array Q.  LDQ >= MAX(1,N).

Z

              Z is a REAL array, dimension (LDZ,N)
              On entry, the leading N-by-N part of
              this array must contain the orthogonal matrix Z from
              the generalized Schur factorization.
              On exit, the leading N-by-N part of this array contains
              the update January 2021
              factorization with integrated eigenvalue reordering.

LDZ

             LDZ is INTEGER
             The leading dimension of the array Z.  LDZ >= MAX(1,N).

NB

            NB is INTEGER
             Block size of the solver planed to use. Typical values
             are 16, 32, or 64. The block size must be an even positive
             integer otherwise an error is returned.

WORK

             WORK is INTEGER array, dimension (LWORK)
             Workspace

LWORK

            LWORK  is INTEGER
            Size of the workspace.
            LWORK >= MAX(1, 4*N, 256)

INFO

             INFO is INTEGER
             == 0:  successful exit;
             < 0:  if INFO = -i, the i-th argument had an illegal
                   value;
             = 1:  The internal STGEX2 caused an error. ]

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 165 of file sla_sort_gev.f90.

Transform the Right-Hand Side or Solution for generalized projections.

Purpose:

 SLA_TRANSFORM_STANDARD computes either

    X <-  QA*X*QB**T                              (1)

 or

    X <- QA**T*X*QB                               (2)

 where QA is a M-by-M matrices, QB is a N-by-N matrix, and X is a M-by-N matrix.

Remarks

The function does not perform any check on the input arguments.

Parameters

TRANS

          TRANS is CHARACTER(1)
          Specifies which transformation is applied:
          == 'N':  Transformation (1)
          == 'T':  Transformation (2)

M

          M is INTEGER
          The order of the matrices QA and the number of rows of X.  M >= 0.

N

          N is INTEGER
          The order of the matrices QB and the number of columns of X.  N >= 0.

X

          X is REAL array, dimension (LDX,N)
          The matrix X is a general M-by-N matrix overwritten by (1) or (2)

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,M).

QA

          QA is REAL array, dimension (LDQA,M)
          The matrix QA is M-by-M the transformation matrix.

LDQA

          LDQA is INTEGER
          The leading dimension of the array QA.  LDQA >= max(1,M).

QB

          QB is REAL array, dimension (LDQB,N)
          The matrix QB is N-by-N the transformation matrix.

LDQB

          LDQB is INTEGER
          The leading dimension of the array QB.  LDQB >= max(1,N).

WORK

          WORK is REAL array, dimension (M*N)
          Workspace for the computation.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 119 of file sla_transform_general.f90.

Transform the Right-Hand Side or Solution for standard projections.

Purpose:

 SLA_TRANSFORM_STANDARD computes either

    X <-  Q*X*Q**T                              (1)

 or

    X <- Q**T*X*Q                               (2)

 where Q and X are M-by-M matrices.

Remarks

The function does not perform any check on the input arguments.

Parameters

TRANS

          TRANS is CHARACTER(1)
          Specifies which transformation is applied:
          == 'N':  Transformation (1)
          == 'T':  Transformation (2)

M

          M is INTEGER
          The order of the matrices Q and X.  M >= 0.

X

          X is REAL array, dimension (LDX,M)
          The matrix X is a general M-by-M matrix overwritten by (1) or (2)

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,M).

Q

          Q is REAL array, dimension (LDQ,M)
          The matrix Q is M-by-M the transformation matrix.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.  LDQ >= max(1,M).

WORK

          WORK is REAL array, dimension (M**2)
          Workspace for the computation.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 101 of file sla_transform_standard.f90.