SUBROUTINE MB02RZ( TRANS, N, NRHS, H, LDH, IPIV, B, LDB, INFO )
C
C PURPOSE
C
C To solve a system of linear equations
C H * X = B, H' * X = B or H**H * X = B
C with a complex upper Hessenberg N-by-N matrix H using the LU
C factorization computed by MB02SZ.
C
C ARGUMENTS
C
C Mode Parameters
C
C TRANS CHARACTER*1
C Specifies the form of the system of equations:
C = 'N': H * X = B (No transpose)
C = 'T': H'* X = B (Transpose)
C = 'C': H**H * X = B (Conjugate transpose)
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the matrix H. N >= 0.
C
C NRHS (input) INTEGER
C The number of right hand sides, i.e., the number of
C columns of the matrix B. NRHS >= 0.
C
C H (input) COMPLEX*16 array, dimension (LDH,N)
C The factors L and U from the factorization H = P*L*U
C as computed by MB02SZ.
C
C LDH INTEGER
C The leading dimension of the array H. LDH >= max(1,N).
C
C IPIV (input) INTEGER array, dimension (N)
C The pivot indices from MB02SZ; for 1<=i<=N, row i of the
C matrix was interchanged with row IPIV(i).
C
C B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
C On entry, the right hand side matrix B.
C On exit, the solution matrix X.
C
C LDB INTEGER
C The leading dimension of the array B. LDB >= max(1,N).
C
C INFO (output) INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C METHOD
C
C The routine uses the factorization
C H = P * L * U
C where P is a permutation matrix, L is lower triangular with unit
C diagonal elements (and one nonzero subdiagonal), and U is upper
C triangular.
C
C REFERENCES
C
C -
C
C NUMERICAL ASPECTS
C 2
C The algorithm requires 0( N x NRHS ) complex operations.
C
C CONTRIBUTOR
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, Dec. 1996.
C Supersedes Release 2.0 routine TB01FW by A.J. Laub, University of
C Southern California, United States of America, May 1980.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Frequency response, Hessenberg form, matrix algebra.
C
C ******************************************************************
C
C .. Parameters ..
COMPLEX*16 ONE
PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
C .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDB, LDH, N, NRHS
C ..
C .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX*16 B( LDB, * ), H( LDH, * )
C .. Local Scalars ..
LOGICAL NOTRAN
INTEGER J, JP
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL XERBLA, ZAXPY, ZSWAP, ZTRSM
C .. Intrinsic Functions ..
INTRINSIC DCONJG, MAX
C .. Executable Statements ..
C
C Test the input parameters.
C
INFO = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'MB02RZ', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
C
IF( NOTRAN ) THEN
C
C Solve H * X = B.
C
C Solve L * X = B, overwriting B with X.
C
C L is represented as a product of permutations and unit lower
C triangular matrices L = P(1) * L(1) * ... * P(n-1) * L(n-1),
C where each transformation L(i) is a rank-one modification of
C the identity matrix.
C
DO 10 J = 1, N - 1
JP = IPIV( J )
IF( JP.NE.J )
$ CALL ZSWAP( NRHS, B( JP, 1 ), LDB, B( J, 1 ), LDB )
CALL ZAXPY( NRHS, -H( J+1, J ), B( J, 1 ), LDB, B( J+1, 1 ),
$ LDB )
10 CONTINUE
C
C Solve U * X = B, overwriting B with X.
C
CALL ZTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N,
$ NRHS, ONE, H, LDH, B, LDB )
C
ELSE IF( LSAME( TRANS, 'T' ) ) THEN
C
C Solve H' * X = B.
C
C Solve U' * X = B, overwriting B with X.
C
CALL ZTRSM( 'Left', 'Upper', TRANS, 'Non-unit', N, NRHS, ONE,
$ H, LDH, B, LDB )
C
C Solve L' * X = B, overwriting B with X.
C
DO 20 J = N - 1, 1, -1
CALL ZAXPY( NRHS, -H( J+1, J ), B( J+1, 1 ), LDB, B( J, 1 ),
$ LDB )
JP = IPIV( J )
IF( JP.NE.J )
$ CALL ZSWAP( NRHS, B( JP, 1 ), LDB, B( J, 1 ), LDB )
20 CONTINUE
C
ELSE
C
C Solve H**H * X = B.
C
C Solve U**H * X = B, overwriting B with X.
C
CALL ZTRSM( 'Left', 'Upper', TRANS, 'Non-unit', N, NRHS, ONE,
$ H, LDH, B, LDB )
C
C Solve L**H * X = B, overwriting B with X.
C
DO 30 J = N - 1, 1, -1
CALL ZAXPY( NRHS, -DCONJG( H( J+1, J ) ), B( J+1, 1 ), LDB,
$ B( J, 1 ), LDB )
JP = IPIV( J )
IF( JP.NE.J )
$ CALL ZSWAP( NRHS, B( JP, 1 ), LDB, B( J, 1 ), LDB )
30 CONTINUE
C
END IF
C
RETURN
C *** Last line of MB02RZ ***
END