SUBROUTINE ZTRSYL( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
$ LDC, SCALE, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER TRANA, TRANB
INTEGER INFO, ISGN, LDA, LDB, LDC, M, N
DOUBLE PRECISION SCALE
* ..
* .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * )
* ..
*
* Purpose
* =======
*
* ZTRSYL solves the complex Sylvester matrix equation:
*
* op(A)*X + X*op(B) = scale*C or
* op(A)*X - X*op(B) = scale*C,
*
* where op(A) = A or A**H, and A and B are both upper triangular. A is
* M-by-M and B is N-by-N; the right hand side C and the solution X are
* M-by-N; and scale is an output scale factor, set <= 1 to avoid
* overflow in X.
*
* Arguments
* =========
*
* TRANA (input) CHARACTER*1
* Specifies the option op(A):
* = 'N': op(A) = A (No transpose)
* = 'C': op(A) = A**H (Conjugate transpose)
*
* TRANB (input) CHARACTER*1
* Specifies the option op(B):
* = 'N': op(B) = B (No transpose)
* = 'C': op(B) = B**H (Conjugate transpose)
*
* ISGN (input) INTEGER
* Specifies the sign in the equation:
* = +1: solve op(A)*X + X*op(B) = scale*C
* = -1: solve op(A)*X - X*op(B) = scale*C
*
* M (input) INTEGER
* The order of the matrix A, and the number of rows in the
* matrices X and C. M >= 0.
*
* N (input) INTEGER
* The order of the matrix B, and the number of columns in the
* matrices X and C. N >= 0.
*
* A (input) COMPLEX*16 array, dimension (LDA,M)
* The upper triangular matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input) COMPLEX*16 array, dimension (LDB,N)
* The upper triangular matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* C (input/output) COMPLEX*16 array, dimension (LDC,N)
* On entry, the M-by-N right hand side matrix C.
* On exit, C is overwritten by the solution matrix X.
*
* LDC (input) INTEGER
* The leading dimension of the array C. LDC >= max(1,M)
*
* SCALE (output) DOUBLE PRECISION
* The scale factor, scale, set <= 1 to avoid overflow in X.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* = 1: A and B have common or very close eigenvalues; perturbed
* values were used to solve the equation (but the matrices
* A and B are unchanged).
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRNA, NOTRNB
INTEGER J, K, L
DOUBLE PRECISION BIGNUM, DA11, DB, EPS, SCALOC, SGN, SMIN,
$ SMLNUM
COMPLEX*16 A11, SUML, SUMR, VEC, X11
* ..
* .. Local Arrays ..
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, ZLANGE
COMPLEX*16 ZDOTC, ZDOTU, ZLADIV
EXTERNAL LSAME, DLAMCH, ZLANGE, ZDOTC, ZDOTU, ZLADIV
* ..
* .. External Subroutines ..
EXTERNAL DLABAD, XERBLA, ZDSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN
* ..
* .. Executable Statements ..
*
* Decode and Test input parameters
*
NOTRNA = LSAME( TRANA, 'N' )
NOTRNB = LSAME( TRANB, 'N' )
*
INFO = 0
IF( .NOT.NOTRNA .AND. .NOT.LSAME( TRANA, 'C' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRNB .AND. .NOT.LSAME( TRANB, 'C' ) ) THEN
INFO = -2
ELSE IF( ISGN.NE.1 .AND. ISGN.NE.-1 ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZTRSYL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Set constants to control overflow
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
CALL DLABAD( SMLNUM, BIGNUM )
SMLNUM = SMLNUM*DBLE( M*N ) / EPS
BIGNUM = ONE / SMLNUM
SMIN = MAX( SMLNUM, EPS*ZLANGE( 'M', M, M, A, LDA, DUM ),
$ EPS*ZLANGE( 'M', N, N, B, LDB, DUM ) )
SCALE = ONE
SGN = ISGN
*
IF( NOTRNA .AND. NOTRNB ) THEN
*
* Solve A*X + ISGN*X*B = scale*C.
*
* The (K,L)th block of X is determined starting from
* bottom-left corner column by column by
*
* A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
*
* Where
* M L-1
* R(K,L) = SUM [A(K,I)*X(I,L)] +ISGN*SUM [X(K,J)*B(J,L)].
* I=K+1 J=1
*
DO 30 L = 1, N
DO 20 K = M, 1, -1
*
SUML = ZDOTU( M-K, A( K, MIN( K+1, M ) ), LDA,
$ C( MIN( K+1, M ), L ), 1 )
SUMR = ZDOTU( L-1, C( K, 1 ), LDC, B( 1, L ), 1 )
VEC = C( K, L ) - ( SUML+SGN*SUMR )
*
SCALOC = ONE
A11 = A( K, K ) + SGN*B( L, L )
DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
$ SCALOC = ONE / DB
END IF
X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
*
IF( SCALOC.NE.ONE ) THEN
DO 10 J = 1, N
CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
10 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K, L ) = X11
*
20 CONTINUE
30 CONTINUE
*
ELSE IF( .NOT.NOTRNA .AND. NOTRNB ) THEN
*
* Solve A' *X + ISGN*X*B = scale*C.
*
* The (K,L)th block of X is determined starting from
* upper-left corner column by column by
*
* A'(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
*
* Where
* K-1 L-1
* R(K,L) = SUM [A'(I,K)*X(I,L)] + ISGN*SUM [X(K,J)*B(J,L)]
* I=1 J=1
*
DO 60 L = 1, N
DO 50 K = 1, M
*
SUML = ZDOTC( K-1, A( 1, K ), 1, C( 1, L ), 1 )
SUMR = ZDOTU( L-1, C( K, 1 ), LDC, B( 1, L ), 1 )
VEC = C( K, L ) - ( SUML+SGN*SUMR )
*
SCALOC = ONE
A11 = DCONJG( A( K, K ) ) + SGN*B( L, L )
DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
$ SCALOC = ONE / DB
END IF
*
X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
*
IF( SCALOC.NE.ONE ) THEN
DO 40 J = 1, N
CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
40 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K, L ) = X11
*
50 CONTINUE
60 CONTINUE
*
ELSE IF( .NOT.NOTRNA .AND. .NOT.NOTRNB ) THEN
*
* Solve A'*X + ISGN*X*B' = C.
*
* The (K,L)th block of X is determined starting from
* upper-right corner column by column by
*
* A'(K,K)*X(K,L) + ISGN*X(K,L)*B'(L,L) = C(K,L) - R(K,L)
*
* Where
* K-1
* R(K,L) = SUM [A'(I,K)*X(I,L)] +
* I=1
* N
* ISGN*SUM [X(K,J)*B'(L,J)].
* J=L+1
*
DO 90 L = N, 1, -1
DO 80 K = 1, M
*
SUML = ZDOTC( K-1, A( 1, K ), 1, C( 1, L ), 1 )
SUMR = ZDOTC( N-L, C( K, MIN( L+1, N ) ), LDC,
$ B( L, MIN( L+1, N ) ), LDB )
VEC = C( K, L ) - ( SUML+SGN*DCONJG( SUMR ) )
*
SCALOC = ONE
A11 = DCONJG( A( K, K )+SGN*B( L, L ) )
DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
$ SCALOC = ONE / DB
END IF
*
X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
*
IF( SCALOC.NE.ONE ) THEN
DO 70 J = 1, N
CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
70 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K, L ) = X11
*
80 CONTINUE
90 CONTINUE
*
ELSE IF( NOTRNA .AND. .NOT.NOTRNB ) THEN
*
* Solve A*X + ISGN*X*B' = C.
*
* The (K,L)th block of X is determined starting from
* bottom-left corner column by column by
*
* A(K,K)*X(K,L) + ISGN*X(K,L)*B'(L,L) = C(K,L) - R(K,L)
*
* Where
* M N
* R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B'(L,J)]
* I=K+1 J=L+1
*
DO 120 L = N, 1, -1
DO 110 K = M, 1, -1
*
SUML = ZDOTU( M-K, A( K, MIN( K+1, M ) ), LDA,
$ C( MIN( K+1, M ), L ), 1 )
SUMR = ZDOTC( N-L, C( K, MIN( L+1, N ) ), LDC,
$ B( L, MIN( L+1, N ) ), LDB )
VEC = C( K, L ) - ( SUML+SGN*DCONJG( SUMR ) )
*
SCALOC = ONE
A11 = A( K, K ) + SGN*DCONJG( B( L, L ) )
DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
$ SCALOC = ONE / DB
END IF
*
X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
*
IF( SCALOC.NE.ONE ) THEN
DO 100 J = 1, N
CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
100 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K, L ) = X11
*
110 CONTINUE
120 CONTINUE
*
END IF
*
RETURN
*
* End of ZTRSYL
*
END