MB02KD

Computation of the product C = alpha op( T ) B + beta C, with T a block Toeplitz matrix

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

  To compute the matrix product

            C = alpha*op( T )*B + beta*C,

  where alpha and beta are scalars and T is a block Toeplitz matrix
  specified by its first block column TC and first block row TR;
  B and C are general matrices of appropriate dimensions.

Specification
      SUBROUTINE MB02KD( LDBLK, TRANS, K, L, M, N, R, ALPHA, BETA,
     $                   TC, LDTC, TR, LDTR, B, LDB, C, LDC, DWORK,
     $                   LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         LDBLK, TRANS
      INTEGER           INFO, K, L, LDB, LDC, LDTC, LDTR, LDWORK, M, N,
     $                  R
      DOUBLE PRECISION  ALPHA, BETA
C     .. Array Arguments ..
      DOUBLE PRECISION  B(LDB,*), C(LDC,*), DWORK(*), TC(LDTC,*),
     $                  TR(LDTR,*)

Arguments

Mode Parameters

  LDBLK   CHARACTER*1
          Specifies where the (1,1)-block of T is stored, as
          follows:
          = 'C':  in the first block of TC;
          = 'R':  in the first block of TR.

  TRANS   CHARACTER*1
          Specifies the form of op( T ) to be used in the matrix
          multiplication as follows:
          = 'N':  op( T ) = T;
          = 'T':  op( T ) = T';
          = 'C':  op( T ) = T'.

Input/Output Parameters
  K       (input) INTEGER
          The number of rows in the blocks of T.  K >= 0.

  L       (input) INTEGER
          The number of columns in the blocks of T.  L >= 0.

  M       (input) INTEGER
          The number of blocks in the first block column of T.
          M >= 0.

  N       (input) INTEGER
          The number of blocks in the first block row of T.  N >= 0.

  R       (input) INTEGER
          The number of columns in B and C.  R >= 0.

  ALPHA   (input) DOUBLE PRECISION
          The scalar alpha. When alpha is zero then TC, TR and B
          are not referenced.

  BETA    (input) DOUBLE PRECISION
          The scalar beta. When beta is zero then C need not be set
          before entry.

  TC      (input)  DOUBLE PRECISION array, dimension (LDTC,L)
          On entry with LDBLK = 'C', the leading M*K-by-L part of
          this array must contain the first block column of T.
          On entry with LDBLK = 'R', the leading (M-1)*K-by-L part
          of this array must contain the 2nd to the M-th blocks of
          the first block column of T.

  LDTC    INTEGER
          The leading dimension of the array TC.
          LDTC >= MAX(1,M*K),      if LDBLK = 'C';
          LDTC >= MAX(1,(M-1)*K),  if LDBLK = 'R'.

  TR      (input)  DOUBLE PRECISION array, dimension (LDTR,k)
          where k is (N-1)*L when LDBLK = 'C' and is N*L when
          LDBLK = 'R'.
          On entry with LDBLK = 'C', the leading K-by-(N-1)*L part
          of this array must contain the 2nd to the N-th blocks of
          the first block row of T.
          On entry with LDBLK = 'R', the leading K-by-N*L part of
          this array must contain the first block row of T.

  LDTR    INTEGER
          The leading dimension of the array TR.  LDTR >= MAX(1,K).

  B       (input)  DOUBLE PRECISION array, dimension (LDB,R)
          On entry with TRANS = 'N', the leading N*L-by-R part of
          this array must contain the matrix B.
          On entry with TRANS = 'T' or TRANS = 'C', the leading
          M*K-by-R part of this array must contain the matrix B.

  LDB     INTEGER
          The leading dimension of the array B.
          LDB >= MAX(1,N*L),  if TRANS = 'N';
          LDB >= MAX(1,M*K),  if TRANS = 'T' or TRANS = 'C'.

  C       (input/output)  DOUBLE PRECISION array, dimension (LDC,R)
          On entry with TRANS = 'N', the leading M*K-by-R part of
          this array must contain the matrix C.
          On entry with TRANS = 'T' or TRANS = 'C', the leading
          N*L-by-R part of this array must contain the matrix C.
          On exit with TRANS = 'N', the leading M*K-by-R part of
          this array contains the updated matrix C.
          On exit with TRANS = 'T' or TRANS = 'C', the leading
          N*L-by-R part of this array contains the updated matrix C.

  LDC     INTEGER
          The leading dimension of the array C.
          LDC >= MAX(1,M*K),  if TRANS = 'N';
          LDC >= MAX(1,N*L),  if TRANS = 'T' or TRANS = 'C'.

Workspace
  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0,  DWORK(1)  returns the optimal
          value of LDWORK.
          On exit, if  INFO = -19,  DWORK(1)  returns the minimum
          value of LDWORK.

  LDWORK  INTEGER
          The length of the array DWORK.  LDWORK >= 1.
          For optimum performance LDWORK should be larger.

          If LDWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          DWORK array, returns this value as the first entry of
          the DWORK array, and no error message related to LDWORK
          is issued by XERBLA.

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value.

Method
  For point Toeplitz matrices or sufficiently large block Toeplitz
  matrices, this algorithm uses convolution algorithms based on
  the fast Hartley transforms [1]. Otherwise, TC is copied in
  reversed order into the workspace such that C can be computed from
  barely M matrix-by-matrix multiplications.

References
  [1] Van Loan, Charles.
      Computational frameworks for the fast Fourier transform.
      SIAM, 1992.

Numerical Aspects
  The algorithm requires O( (K*L+R*L+K*R)*(N+M)*log(N+M) + K*L*R )
  floating point operations.

Further Comments
  None
Example

Program Text

*     MB02KD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
      DOUBLE PRECISION  ZERO, ONE
      PARAMETER         ( ZERO = 0.0D0, ONE = 1.0D0 )
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          KMAX, LMAX, MMAX, NMAX, RMAX
      PARAMETER        ( KMAX = 20, LMAX = 20, MMAX = 20, NMAX = 20,
     $                   RMAX = 20 )
      INTEGER          LDB, LDC, LDTC, LDTR, LDWORK
      PARAMETER        ( LDB  = LMAX*NMAX, LDC  = KMAX*MMAX,
     $                   LDTC = MMAX*KMAX, LDTR = KMAX,
     $                   LDWORK = 2*( KMAX*LMAX + KMAX*RMAX
     $                            + LMAX*RMAX + 1 )*( MMAX + NMAX ) )
*     .. Local Scalars ..
      INTEGER          I, INFO, J, K, L, M, N, R
      CHARACTER        LDBLK, TRANS
      DOUBLE PRECISION ALPHA, BETA
*     .. Local Arrays .. (Dimensioned for TRANS = 'N'.)
      DOUBLE PRECISION B(LDB,RMAX), C(LDC,RMAX), DWORK(LDWORK),
     $                 TC(LDTC,LMAX), TR(LDTR,NMAX*LMAX)
*     .. External Functions ..
      LOGICAL          LSAME
      EXTERNAL         LSAME
*     .. External Subroutines ..
      EXTERNAL         MB02KD
*     .. Intrinsic Functions ..
      INTRINSIC        MAX
*
*     .. Executable Statements ..
      WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
      READ ( NIN, FMT = '()' )
      READ ( NIN, FMT = * )  K, L, M, N, R, LDBLK, TRANS
      IF( K.LE.0 .OR. K.GT.KMAX ) THEN
         WRITE ( NOUT, FMT = 99994 ) K
      ELSE IF( L.LE.0 .OR. L.GT.LMAX ) THEN
         WRITE ( NOUT, FMT = 99993 ) L
      ELSE IF( M.LE.0 .OR. M.GT.MMAX ) THEN
         WRITE ( NOUT, FMT = 99992 ) M
      ELSE IF( N.LE.0 .OR. N.GT.NMAX ) THEN
         WRITE ( NOUT, FMT = 99991 ) N
      ELSE IF( R.LE.0 .OR. R.GT.RMAX ) THEN
         WRITE ( NOUT, FMT = 99990 ) N
      ELSE
         IF ( LSAME( LDBLK, 'R' ) ) THEN
            READ ( NIN, FMT = * ) ( ( TC(I,J), J = 1,L ),
     $                              I = 1,(M-1)*K )
            READ ( NIN, FMT = * ) ( ( TR(I,J), J = 1,N*L ), I = 1,K )
         ELSE
            READ ( NIN, FMT = * ) ( ( TC(I,J), J = 1,L ), I = 1,M*K )
            READ ( NIN, FMT = * ) ( ( TR(I,J), J = 1,(N-1)*L ),
     $                              I = 1,K )
         END IF
         IF ( LSAME( TRANS, 'N' ) ) THEN
            READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,R ), I = 1,N*L )
         ELSE
            READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,R ), I = 1,M*K )
         END IF
         ALPHA = ONE
         BETA  = ZERO
         CALL MB02KD( LDBLK, TRANS, K, L, M, N, R, ALPHA, BETA, TC,
     $                LDTC, TR, LDTR, B, LDB, C, LDC, DWORK, LDWORK,
     $                INFO )
         IF ( INFO.NE.0 ) THEN
            WRITE ( NOUT, FMT = 99998 ) INFO
         ELSE
            IF ( LSAME( TRANS, 'N' ) ) THEN
               WRITE ( NOUT, FMT = 99997 )
               DO 10  I = 1, M*K
                  WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,R )
   10          CONTINUE
            ELSE
               WRITE ( NOUT, FMT = 99996 )
               DO 20  I = 1, N*L
                  WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,R )
   20          CONTINUE
            END IF
         END IF
      END IF
      STOP
*
99999 FORMAT (' MB02KD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02KD = ',I2)
99997 FORMAT (' The product C = T * B is ')
99996 FORMAT (' The product C = T^T * B is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' K is out of range.',/' K = ',I5)
99993 FORMAT (/' L is out of range.',/' L = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' R is out of range.',/' R = ',I5)
      END
Program Data
MB02KD EXAMPLE PROGRAM DATA
   3    2   4    5    1    C    N
     4.0     1.0
     3.0     5.0
     2.0     1.0
     4.0     1.0
     3.0     4.0
     2.0     4.0
     3.0     1.0
     3.0     0.0
     4.0     4.0
     5.0     1.0
     3.0     1.0
     4.0     3.0
     5.0     2.0     2.0     2.0     2.0     1.0     1.0     3.0
     4.0     1.0     5.0     4.0     5.0     4.0     1.0     2.0
     2.0     3.0     4.0     1.0     3.0     3.0     3.0     3.0
     0.0
     2.0
     2.0
     2.0
     1.0
     3.0
     3.0
     4.0
     2.0
     3.0
Program Results
 MB02KD EXAMPLE PROGRAM RESULTS

 The product C = T * B is 
  45.0000
  76.0000
  55.0000
  44.0000
  84.0000
  56.0000
  52.0000
  70.0000
  54.0000
  49.0000
  63.0000
  59.0000

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