SUBROUTINE SB04RY( RC, UL, M, A, LDA, LAMBDA, D, TOL, IWORK, $ DWORK, LDDWOR, INFO ) C C RELEASE 4.0, WGS COPYRIGHT 2000. C C PURPOSE C C To solve a system of equations in Hessenberg form with one C right-hand side. C C ARGUMENTS C C Mode Parameters C C RC CHARACTER*1 C Indicates processing by columns or rows, as follows: C = 'R': Row transformations are applied; C = 'C': Column transformations are applied. C C UL CHARACTER*1 C Indicates whether A is upper or lower Hessenberg matrix, C as follows: C = 'U': A is upper Hessenberg; C = 'L': A is lower Hessenberg. C C Input/Output Parameters C C M (input) INTEGER C The order of the matrix A. M >= 0. C C A (input) DOUBLE PRECISION array, dimension (LDA,M) C The leading M-by-M part of this array must contain a C matrix A in Hessenberg form. C C LDA INTEGER C The leading dimension of array A. LDA >= MAX(1,M). C C LAMBDA (input) DOUBLE PRECISION C This variable must contain the value to be multiplied with C the elements of A. C C D (input/output) DOUBLE PRECISION array, dimension (M) C On entry, this array must contain the right-hand side C vector of the Hessenberg system. C On exit, if INFO = 0, this array contains the solution C vector of the Hessenberg system. C C Tolerances C C TOL DOUBLE PRECISION C The tolerance to be used to test for near singularity of C the triangular factor R of the Hessenberg matrix. A matrix C whose estimated condition number is less than 1/TOL is C considered to be nonsingular. C C Workspace C C IWORK INTEGER array, dimension (M) C C DWORK DOUBLE PRECISION array, dimension (LDDWOR,M+3) C The leading M-by-M part of this array is used for C computing the triangular factor of the QR decomposition C of the Hessenberg matrix. The remaining 3*M elements are C used as workspace for the computation of the reciprocal C condition estimate. C C LDDWOR INTEGER C The leading dimension of array DWORK. LDDWOR >= MAX(1,M). C C Error Indicator C C INFO INTEGER C = 0: successful exit; C = 1: if the Hessenberg matrix is (numerically) singular. C That is, its estimated reciprocal condition number C is less than or equal to TOL. C C NUMERICAL ASPECTS C C None. C C CONTRIBUTORS C C D. Sima, University of Bucharest, May 2000. C C REVISIONS C C - C C Note that RC, UL, M, LDA, and LDDWOR must be such that the value C of the LOGICAL variable OK in the following statement is true. C C OK = ( ( UL.EQ.'U' ) .OR. ( UL.EQ.'u' ) .OR. C ( UL.EQ.'L' ) .OR. ( UL.EQ.'l' ) ) C .AND. C ( ( RC.EQ.'R' ) .OR. ( RC.EQ.'r' ) .OR. C ( RC.EQ.'C' ) .OR. ( RC.EQ.'c' ) ) C .AND. C ( M.GE.0 ) C .AND. C ( LDA.GE.MAX( 1, M ) ) C .AND. C ( LDDWOR.GE.MAX( 1, M ) ) C C These conditions are not checked by the routine. C C KEYWORDS C C Hessenberg form, orthogonal transformation, real Schur form, C Sylvester equation. C C ****************************************************************** C DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 ) C .. Scalar Arguments .. CHARACTER RC, UL INTEGER INFO, LDA, LDDWOR, M DOUBLE PRECISION LAMBDA, TOL C .. Array Arguments .. INTEGER IWORK(*) DOUBLE PRECISION A(LDA,*), D(*), DWORK(LDDWOR,*) C .. Local Scalars .. CHARACTER TRANS INTEGER J, J1, MJ DOUBLE PRECISION C, R, RCOND, S C .. External Functions .. LOGICAL LSAME EXTERNAL LSAME C .. External Subroutines .. EXTERNAL DCOPY, DLARTG, DROT, DSCAL, DTRCON, DTRSV C .. Intrinsic Functions .. INTRINSIC MAX, MIN C .. Executable Statements .. C INFO = 0 C C For speed, no tests on the input scalar arguments are made. C Quick return if possible. C IF ( M.EQ.0 ) $ RETURN C IF ( LSAME( UL, 'U' ) ) THEN C DO 20 J = 1, M CALL DCOPY( MIN( J+1, M ), A(1,J), 1, DWORK(1,J), 1 ) CALL DSCAL( MIN( J+1, M ), LAMBDA, DWORK(1,J), 1 ) DWORK(J,J) = DWORK(J,J) + ONE 20 CONTINUE C IF ( LSAME( RC, 'R' ) ) THEN TRANS = 'N' C C A is an upper Hessenberg matrix, row transformations. C DO 40 J = 1, M - 1 MJ = M - J IF ( DWORK(J+1,J).NE.ZERO ) THEN CALL DLARTG( DWORK(J,J), DWORK(J+1,J), C, S, R ) DWORK(J,J) = R DWORK(J+1,J) = ZERO CALL DROT( MJ, DWORK(J,J+1), LDDWOR, DWORK(J+1,J+1), $ LDDWOR, C, S ) CALL DROT( 1, D(J), 1, D(J+1), 1, C, S ) END IF 40 CONTINUE C ELSE TRANS = 'T' C C A is an upper Hessenberg matrix, column transformations. C DO 60 J = 1, M - 1 MJ = M - J IF ( DWORK(MJ+1,MJ).NE.ZERO ) THEN CALL DLARTG( DWORK(MJ+1,MJ+1), DWORK(MJ+1,MJ), C, S, $ R ) DWORK(MJ+1,MJ+1) = R DWORK(MJ+1,MJ) = ZERO CALL DROT( MJ, DWORK(1,MJ+1), 1, DWORK(1,MJ), 1, C, $ S ) CALL DROT( 1, D(MJ+1), 1, D(MJ), 1, C, S ) END IF 60 CONTINUE C END IF ELSE C DO 80 J = 1, M J1 = MAX( J - 1, 1 ) CALL DCOPY( M-J1+1, A(J1,J), 1, DWORK(J1,J), 1 ) CALL DSCAL( M-J1+1, LAMBDA, DWORK(J1,J), 1 ) DWORK(J,J) = DWORK(J,J) + ONE 80 CONTINUE C IF ( LSAME( RC, 'R' ) ) THEN TRANS = 'N' C C A is a lower Hessenberg matrix, row transformations. C DO 100 J = 1, M - 1 MJ = M - J IF ( DWORK(MJ,MJ+1).NE.ZERO ) THEN CALL DLARTG( DWORK(MJ+1,MJ+1), DWORK(MJ,MJ+1), C, S, $ R ) DWORK(MJ+1,MJ+1) = R DWORK(MJ,MJ+1) = ZERO CALL DROT( MJ, DWORK(MJ+1,1), LDDWOR, DWORK(MJ,1), $ LDDWOR, C, S ) CALL DROT( 1, D(MJ+1), 1, D(MJ), 1, C, S ) END IF 100 CONTINUE C ELSE TRANS = 'T' C C A is a lower Hessenberg matrix, column transformations. C DO 120 J = 1, M - 1 MJ = M - J IF ( DWORK(J,J+1).NE.ZERO ) THEN CALL DLARTG( DWORK(J,J), DWORK(J,J+1), C, S, R ) DWORK(J,J) = R DWORK(J,J+1) = ZERO CALL DROT( MJ, DWORK(J+1,J), 1, DWORK(J+1,J+1), 1, C, $ S ) CALL DROT( 1, D(J), 1, D(J+1), 1, C, S ) END IF 120 CONTINUE C END IF END IF C CALL DTRCON( '1-norm', UL, 'Non-unit', M, DWORK, LDDWOR, RCOND, $ DWORK(1,M+1), IWORK, INFO ) IF ( RCOND.LE.TOL ) THEN INFO = 1 ELSE CALL DTRSV( UL, TRANS, 'Non-unit', M, DWORK, LDDWOR, D, 1 ) END IF C RETURN C *** Last line of SB04RY *** END