**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.

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,*)

**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'.

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'.

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.

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

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.

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

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

None

**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

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

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