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Diffstat (limited to 'SRC/dgeevx.f')
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diff --git a/SRC/dgeevx.f b/SRC/dgeevx.f new file mode 100644 index 00000000..7d927ae9 --- /dev/null +++ b/SRC/dgeevx.f @@ -0,0 +1,556 @@ + SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, + $ VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, + $ RCONDE, RCONDV, WORK, LWORK, IWORK, INFO ) +* +* -- LAPACK driver routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* .. Scalar Arguments .. + CHARACTER BALANC, JOBVL, JOBVR, SENSE + INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N + DOUBLE PRECISION ABNRM +* .. +* .. Array Arguments .. + INTEGER IWORK( * ) + DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ), + $ SCALE( * ), VL( LDVL, * ), VR( LDVR, * ), + $ WI( * ), WORK( * ), WR( * ) +* .. +* +* Purpose +* ======= +* +* DGEEVX computes for an N-by-N real nonsymmetric matrix A, the +* eigenvalues and, optionally, the left and/or right eigenvectors. +* +* Optionally also, it computes a balancing transformation to improve +* the conditioning of the eigenvalues and eigenvectors (ILO, IHI, +* SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues +* (RCONDE), and reciprocal condition numbers for the right +* eigenvectors (RCONDV). +* +* The right eigenvector v(j) of A satisfies +* A * v(j) = lambda(j) * v(j) +* where lambda(j) is its eigenvalue. +* The left eigenvector u(j) of A satisfies +* u(j)**H * A = lambda(j) * u(j)**H +* where u(j)**H denotes the conjugate transpose of u(j). +* +* The computed eigenvectors are normalized to have Euclidean norm +* equal to 1 and largest component real. +* +* Balancing a matrix means permuting the rows and columns to make it +* more nearly upper triangular, and applying a diagonal similarity +* transformation D * A * D**(-1), where D is a diagonal matrix, to +* make its rows and columns closer in norm and the condition numbers +* of its eigenvalues and eigenvectors smaller. The computed +* reciprocal condition numbers correspond to the balanced matrix. +* Permuting rows and columns will not change the condition numbers +* (in exact arithmetic) but diagonal scaling will. For further +* explanation of balancing, see section 4.10.2 of the LAPACK +* Users' Guide. +* +* Arguments +* ========= +* +* BALANC (input) CHARACTER*1 +* Indicates how the input matrix should be diagonally scaled +* and/or permuted to improve the conditioning of its +* eigenvalues. +* = 'N': Do not diagonally scale or permute; +* = 'P': Perform permutations to make the matrix more nearly +* upper triangular. Do not diagonally scale; +* = 'S': Diagonally scale the matrix, i.e. replace A by +* D*A*D**(-1), where D is a diagonal matrix chosen +* to make the rows and columns of A more equal in +* norm. Do not permute; +* = 'B': Both diagonally scale and permute A. +* +* Computed reciprocal condition numbers will be for the matrix +* after balancing and/or permuting. Permuting does not change +* condition numbers (in exact arithmetic), but balancing does. +* +* JOBVL (input) CHARACTER*1 +* = 'N': left eigenvectors of A are not computed; +* = 'V': left eigenvectors of A are computed. +* If SENSE = 'E' or 'B', JOBVL must = 'V'. +* +* JOBVR (input) CHARACTER*1 +* = 'N': right eigenvectors of A are not computed; +* = 'V': right eigenvectors of A are computed. +* If SENSE = 'E' or 'B', JOBVR must = 'V'. +* +* SENSE (input) CHARACTER*1 +* Determines which reciprocal condition numbers are computed. +* = 'N': None are computed; +* = 'E': Computed for eigenvalues only; +* = 'V': Computed for right eigenvectors only; +* = 'B': Computed for eigenvalues and right eigenvectors. +* +* If SENSE = 'E' or 'B', both left and right eigenvectors +* must also be computed (JOBVL = 'V' and JOBVR = 'V'). +* +* N (input) INTEGER +* The order of the matrix A. N >= 0. +* +* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) +* On entry, the N-by-N matrix A. +* On exit, A has been overwritten. If JOBVL = 'V' or +* JOBVR = 'V', A contains the real Schur form of the balanced +* version of the input matrix A. +* +* LDA (input) INTEGER +* The leading dimension of the array A. LDA >= max(1,N). +* +* WR (output) DOUBLE PRECISION array, dimension (N) +* WI (output) DOUBLE PRECISION array, dimension (N) +* WR and WI contain the real and imaginary parts, +* respectively, of the computed eigenvalues. Complex +* conjugate pairs of eigenvalues will appear consecutively +* with the eigenvalue having the positive imaginary part +* first. +* +* VL (output) DOUBLE PRECISION array, dimension (LDVL,N) +* If JOBVL = 'V', the left eigenvectors u(j) are stored one +* after another in the columns of VL, in the same order +* as their eigenvalues. +* If JOBVL = 'N', VL is not referenced. +* If the j-th eigenvalue is real, then u(j) = VL(:,j), +* the j-th column of VL. +* If the j-th and (j+1)-st eigenvalues form a complex +* conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and +* u(j+1) = VL(:,j) - i*VL(:,j+1). +* +* LDVL (input) INTEGER +* The leading dimension of the array VL. LDVL >= 1; if +* JOBVL = 'V', LDVL >= N. +* +* VR (output) DOUBLE PRECISION array, dimension (LDVR,N) +* If JOBVR = 'V', the right eigenvectors v(j) are stored one +* after another in the columns of VR, in the same order +* as their eigenvalues. +* If JOBVR = 'N', VR is not referenced. +* If the j-th eigenvalue is real, then v(j) = VR(:,j), +* the j-th column of VR. +* If the j-th and (j+1)-st eigenvalues form a complex +* conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and +* v(j+1) = VR(:,j) - i*VR(:,j+1). +* +* LDVR (input) INTEGER +* The leading dimension of the array VR. LDVR >= 1, and if +* JOBVR = 'V', LDVR >= N. +* +* ILO (output) INTEGER +* IHI (output) INTEGER +* ILO and IHI are integer values determined when A was +* balanced. The balanced A(i,j) = 0 if I > J and +* J = 1,...,ILO-1 or I = IHI+1,...,N. +* +* SCALE (output) DOUBLE PRECISION array, dimension (N) +* Details of the permutations and scaling factors applied +* when balancing A. If P(j) is the index of the row and column +* interchanged with row and column j, and D(j) is the scaling +* factor applied to row and column j, then +* SCALE(J) = P(J), for J = 1,...,ILO-1 +* = D(J), for J = ILO,...,IHI +* = P(J) for J = IHI+1,...,N. +* The order in which the interchanges are made is N to IHI+1, +* then 1 to ILO-1. +* +* ABNRM (output) DOUBLE PRECISION +* The one-norm of the balanced matrix (the maximum +* of the sum of absolute values of elements of any column). +* +* RCONDE (output) DOUBLE PRECISION array, dimension (N) +* RCONDE(j) is the reciprocal condition number of the j-th +* eigenvalue. +* +* RCONDV (output) DOUBLE PRECISION array, dimension (N) +* RCONDV(j) is the reciprocal condition number of the j-th +* right eigenvector. +* +* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) +* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. +* +* LWORK (input) INTEGER +* The dimension of the array WORK. If SENSE = 'N' or 'E', +* LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V', +* LWORK >= 3*N. If SENSE = 'V' or 'B', LWORK >= N*(N+6). +* For good performance, LWORK must generally be larger. +* +* If LWORK = -1, then a workspace query is assumed; the routine +* only calculates the optimal size of the WORK array, returns +* this value as the first entry of the WORK array, and no error +* message related to LWORK is issued by XERBLA. +* +* IWORK (workspace) INTEGER array, dimension (2*N-2) +* If SENSE = 'N' or 'E', not referenced. +* +* INFO (output) INTEGER +* = 0: successful exit +* < 0: if INFO = -i, the i-th argument had an illegal value. +* > 0: if INFO = i, the QR algorithm failed to compute all the +* eigenvalues, and no eigenvectors or condition numbers +* have been computed; elements 1:ILO-1 and i+1:N of WR +* and WI contain eigenvalues which have converged. +* +* ===================================================================== +* +* .. Parameters .. + DOUBLE PRECISION ZERO, ONE + PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) +* .. +* .. Local Scalars .. + LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE, + $ WNTSNN, WNTSNV + CHARACTER JOB, SIDE + INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K, MAXWRK, + $ MINWRK, NOUT + DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM, + $ SN +* .. +* .. Local Arrays .. + LOGICAL SELECT( 1 ) + DOUBLE PRECISION DUM( 1 ) +* .. +* .. External Subroutines .. + EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY, + $ DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC, + $ DTRSNA, XERBLA +* .. +* .. External Functions .. + LOGICAL LSAME + INTEGER IDAMAX, ILAENV + DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2 + EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2, + $ DNRM2 +* .. +* .. Intrinsic Functions .. + INTRINSIC MAX, SQRT +* .. +* .. Executable Statements .. +* +* Test the input arguments +* + INFO = 0 + LQUERY = ( LWORK.EQ.-1 ) + WANTVL = LSAME( JOBVL, 'V' ) + WANTVR = LSAME( JOBVR, 'V' ) + WNTSNN = LSAME( SENSE, 'N' ) + WNTSNE = LSAME( SENSE, 'E' ) + WNTSNV = LSAME( SENSE, 'V' ) + WNTSNB = LSAME( SENSE, 'B' ) + IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, + $ 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) + $ THEN + INFO = -1 + ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN + INFO = -2 + ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN + INFO = -3 + ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR. + $ ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND. + $ WANTVR ) ) ) THEN + INFO = -4 + ELSE IF( N.LT.0 ) THEN + INFO = -5 + ELSE IF( LDA.LT.MAX( 1, N ) ) THEN + INFO = -7 + ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN + INFO = -11 + ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN + INFO = -13 + END IF +* +* Compute workspace +* (Note: Comments in the code beginning "Workspace:" describe the +* minimal amount of workspace needed at that point in the code, +* as well as the preferred amount for good performance. +* NB refers to the optimal block size for the immediately +* following subroutine, as returned by ILAENV. +* HSWORK refers to the workspace preferred by DHSEQR, as +* calculated below. HSWORK is computed assuming ILO=1 and IHI=N, +* the worst case.) +* + IF( INFO.EQ.0 ) THEN + IF( N.EQ.0 ) THEN + MINWRK = 1 + MAXWRK = 1 + ELSE + MAXWRK = N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 ) +* + IF( WANTVL ) THEN + CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL, + $ WORK, -1, INFO ) + ELSE IF( WANTVR ) THEN + CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR, + $ WORK, -1, INFO ) + ELSE + IF( WNTSNN ) THEN + CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, + $ LDVR, WORK, -1, INFO ) + ELSE + CALL DHSEQR( 'S', 'N', N, 1, N, A, LDA, WR, WI, VR, + $ LDVR, WORK, -1, INFO ) + END IF + END IF + HSWORK = WORK( 1 ) +* + IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN + MINWRK = 2*N + IF( .NOT.WNTSNN ) + $ MINWRK = MAX( MINWRK, N*N+6*N ) + MAXWRK = MAX( MAXWRK, HSWORK ) + IF( .NOT.WNTSNN ) + $ MAXWRK = MAX( MAXWRK, N*N + 6*N ) + ELSE + MINWRK = 3*N + IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) ) + $ MINWRK = MAX( MINWRK, N*N + 6*N ) + MAXWRK = MAX( MAXWRK, HSWORK ) + MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'DORGHR', + $ ' ', N, 1, N, -1 ) ) + IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) ) + $ MAXWRK = MAX( MAXWRK, N*N + 6*N ) + MAXWRK = MAX( MAXWRK, 3*N ) + END IF + MAXWRK = MAX( MAXWRK, MINWRK ) + END IF + WORK( 1 ) = MAXWRK +* + IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN + INFO = -21 + END IF + END IF +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'DGEEVX', -INFO ) + RETURN + ELSE IF( LQUERY ) THEN + RETURN + END IF +* +* Quick return if possible +* + IF( N.EQ.0 ) + $ RETURN +* +* Get machine constants +* + EPS = DLAMCH( 'P' ) + SMLNUM = DLAMCH( 'S' ) + BIGNUM = ONE / SMLNUM + CALL DLABAD( SMLNUM, BIGNUM ) + SMLNUM = SQRT( SMLNUM ) / EPS + BIGNUM = ONE / SMLNUM +* +* Scale A if max element outside range [SMLNUM,BIGNUM] +* + ICOND = 0 + ANRM = DLANGE( 'M', N, N, A, LDA, DUM ) + SCALEA = .FALSE. + IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN + SCALEA = .TRUE. + CSCALE = SMLNUM + ELSE IF( ANRM.GT.BIGNUM ) THEN + SCALEA = .TRUE. + CSCALE = BIGNUM + END IF + IF( SCALEA ) + $ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) +* +* Balance the matrix and compute ABNRM +* + CALL DGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR ) + ABNRM = DLANGE( '1', N, N, A, LDA, DUM ) + IF( SCALEA ) THEN + DUM( 1 ) = ABNRM + CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR ) + ABNRM = DUM( 1 ) + END IF +* +* Reduce to upper Hessenberg form +* (Workspace: need 2*N, prefer N+N*NB) +* + ITAU = 1 + IWRK = ITAU + N + CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), + $ LWORK-IWRK+1, IERR ) +* + IF( WANTVL ) THEN +* +* Want left eigenvectors +* Copy Householder vectors to VL +* + SIDE = 'L' + CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL ) +* +* Generate orthogonal matrix in VL +* (Workspace: need 2*N-1, prefer N+(N-1)*NB) +* + CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ), + $ LWORK-IWRK+1, IERR ) +* +* Perform QR iteration, accumulating Schur vectors in VL +* (Workspace: need 1, prefer HSWORK (see comments) ) +* + IWRK = ITAU + CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL, + $ WORK( IWRK ), LWORK-IWRK+1, INFO ) +* + IF( WANTVR ) THEN +* +* Want left and right eigenvectors +* Copy Schur vectors to VR +* + SIDE = 'B' + CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) + END IF +* + ELSE IF( WANTVR ) THEN +* +* Want right eigenvectors +* Copy Householder vectors to VR +* + SIDE = 'R' + CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR ) +* +* Generate orthogonal matrix in VR +* (Workspace: need 2*N-1, prefer N+(N-1)*NB) +* + CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ), + $ LWORK-IWRK+1, IERR ) +* +* Perform QR iteration, accumulating Schur vectors in VR +* (Workspace: need 1, prefer HSWORK (see comments) ) +* + IWRK = ITAU + CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR, + $ WORK( IWRK ), LWORK-IWRK+1, INFO ) +* + ELSE +* +* Compute eigenvalues only +* If condition numbers desired, compute Schur form +* + IF( WNTSNN ) THEN + JOB = 'E' + ELSE + JOB = 'S' + END IF +* +* (Workspace: need 1, prefer HSWORK (see comments) ) +* + IWRK = ITAU + CALL DHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR, + $ WORK( IWRK ), LWORK-IWRK+1, INFO ) + END IF +* +* If INFO > 0 from DHSEQR, then quit +* + IF( INFO.GT.0 ) + $ GO TO 50 +* + IF( WANTVL .OR. WANTVR ) THEN +* +* Compute left and/or right eigenvectors +* (Workspace: need 3*N) +* + CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, + $ N, NOUT, WORK( IWRK ), IERR ) + END IF +* +* Compute condition numbers if desired +* (Workspace: need N*N+6*N unless SENSE = 'E') +* + IF( .NOT.WNTSNN ) THEN + CALL DTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, + $ RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, IWORK, + $ ICOND ) + END IF +* + IF( WANTVL ) THEN +* +* Undo balancing of left eigenvectors +* + CALL DGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL, + $ IERR ) +* +* Normalize left eigenvectors and make largest component real +* + DO 20 I = 1, N + IF( WI( I ).EQ.ZERO ) THEN + SCL = ONE / DNRM2( N, VL( 1, I ), 1 ) + CALL DSCAL( N, SCL, VL( 1, I ), 1 ) + ELSE IF( WI( I ).GT.ZERO ) THEN + SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ), + $ DNRM2( N, VL( 1, I+1 ), 1 ) ) + CALL DSCAL( N, SCL, VL( 1, I ), 1 ) + CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 ) + DO 10 K = 1, N + WORK( K ) = VL( K, I )**2 + VL( K, I+1 )**2 + 10 CONTINUE + K = IDAMAX( N, WORK, 1 ) + CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R ) + CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN ) + VL( K, I+1 ) = ZERO + END IF + 20 CONTINUE + END IF +* + IF( WANTVR ) THEN +* +* Undo balancing of right eigenvectors +* + CALL DGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR, + $ IERR ) +* +* Normalize right eigenvectors and make largest component real +* + DO 40 I = 1, N + IF( WI( I ).EQ.ZERO ) THEN + SCL = ONE / DNRM2( N, VR( 1, I ), 1 ) + CALL DSCAL( N, SCL, VR( 1, I ), 1 ) + ELSE IF( WI( I ).GT.ZERO ) THEN + SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ), + $ DNRM2( N, VR( 1, I+1 ), 1 ) ) + CALL DSCAL( N, SCL, VR( 1, I ), 1 ) + CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 ) + DO 30 K = 1, N + WORK( K ) = VR( K, I )**2 + VR( K, I+1 )**2 + 30 CONTINUE + K = IDAMAX( N, WORK, 1 ) + CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R ) + CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN ) + VR( K, I+1 ) = ZERO + END IF + 40 CONTINUE + END IF +* +* Undo scaling if necessary +* + 50 CONTINUE + IF( SCALEA ) THEN + CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ), + $ MAX( N-INFO, 1 ), IERR ) + CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ), + $ MAX( N-INFO, 1 ), IERR ) + IF( INFO.EQ.0 ) THEN + IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 ) + $ CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N, + $ IERR ) + ELSE + CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N, + $ IERR ) + CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N, + $ IERR ) + END IF + END IF +* + WORK( 1 ) = MAXWRK + RETURN +* +* End of DGEEVX +* + END |