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SUBROUTINE SGEEVX( 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 3.3.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2011 -- * * .. Scalar Arguments .. CHARACTER BALANC, JOBVL, JOBVR, SENSE INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N REAL ABNRM * .. * .. Array Arguments .. INTEGER IWORK( * ) REAL A( LDA, * ), RCONDE( * ), RCONDV( * ), $ SCALE( * ), VL( LDVL, * ), VR( LDVR, * ), $ WI( * ), WORK( * ), WR( * ) * .. * * Purpose * ======= * * SGEEVX 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)**T * A = lambda(j) * u(j)**T * where u(j)**T denotes the 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) REAL 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) REAL array, dimension (N) * WI (output) REAL 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) REAL 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) REAL 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) REAL 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) REAL * The one-norm of the balanced matrix (the maximum * of the sum of absolute values of elements of any column). * * RCONDE (output) REAL array, dimension (N) * RCONDE(j) is the reciprocal condition number of the j-th * eigenvalue. * * RCONDV (output) REAL array, dimension (N) * RCONDV(j) is the reciprocal condition number of the j-th * right eigenvector. * * WORK (workspace/output) REAL 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 .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) * .. * .. 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 REAL ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM, $ SN * .. * .. Local Arrays .. LOGICAL SELECT( 1 ) REAL DUM( 1 ) * .. * .. External Subroutines .. EXTERNAL SGEBAK, SGEBAL, SGEHRD, SHSEQR, SLABAD, SLACPY, $ SLARTG, SLASCL, SORGHR, SROT, SSCAL, STREVC, $ STRSNA, XERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV, ISAMAX REAL SLAMCH, SLANGE, SLAPY2, SNRM2 EXTERNAL LSAME, ILAENV, ISAMAX, SLAMCH, SLANGE, SLAPY2, $ SNRM2 * .. * .. 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 SHSEQR, 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, 'SGEHRD', ' ', N, 1, N, 0 ) * IF( WANTVL ) THEN CALL SHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL, $ WORK, -1, INFO ) ELSE IF( WANTVR ) THEN CALL SHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR, $ WORK, -1, INFO ) ELSE IF( WNTSNN ) THEN CALL SHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, $ LDVR, WORK, -1, INFO ) ELSE CALL SHSEQR( '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, 'SORGHR', $ ' ', 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( 'SGEEVX', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * Get machine constants * EPS = SLAMCH( 'P' ) SMLNUM = SLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL SLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM * * Scale A if max element outside range [SMLNUM,BIGNUM] * ICOND = 0 ANRM = SLANGE( '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 SLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) * * Balance the matrix and compute ABNRM * CALL SGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR ) ABNRM = SLANGE( '1', N, N, A, LDA, DUM ) IF( SCALEA ) THEN DUM( 1 ) = ABNRM CALL SLASCL( '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 SGEHRD( 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 SLACPY( 'L', N, N, A, LDA, VL, LDVL ) * * Generate orthogonal matrix in VL * (Workspace: need 2*N-1, prefer N+(N-1)*NB) * CALL SORGHR( 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 SHSEQR( '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 SLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) END IF * ELSE IF( WANTVR ) THEN * * Want right eigenvectors * Copy Householder vectors to VR * SIDE = 'R' CALL SLACPY( 'L', N, N, A, LDA, VR, LDVR ) * * Generate orthogonal matrix in VR * (Workspace: need 2*N-1, prefer N+(N-1)*NB) * CALL SORGHR( 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 SHSEQR( '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 SHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR, $ WORK( IWRK ), LWORK-IWRK+1, INFO ) END IF * * If INFO > 0 from SHSEQR, 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 STREVC( 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 STRSNA( 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 SGEBAK( 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 / SNRM2( N, VL( 1, I ), 1 ) CALL SSCAL( N, SCL, VL( 1, I ), 1 ) ELSE IF( WI( I ).GT.ZERO ) THEN SCL = ONE / SLAPY2( SNRM2( N, VL( 1, I ), 1 ), $ SNRM2( N, VL( 1, I+1 ), 1 ) ) CALL SSCAL( N, SCL, VL( 1, I ), 1 ) CALL SSCAL( 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 = ISAMAX( N, WORK, 1 ) CALL SLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R ) CALL SROT( 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 SGEBAK( 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 / SNRM2( N, VR( 1, I ), 1 ) CALL SSCAL( N, SCL, VR( 1, I ), 1 ) ELSE IF( WI( I ).GT.ZERO ) THEN SCL = ONE / SLAPY2( SNRM2( N, VR( 1, I ), 1 ), $ SNRM2( N, VR( 1, I+1 ), 1 ) ) CALL SSCAL( N, SCL, VR( 1, I ), 1 ) CALL SSCAL( 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 = ISAMAX( N, WORK, 1 ) CALL SLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R ) CALL SROT( 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 SLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ), $ MAX( N-INFO, 1 ), IERR ) CALL SLASCL( '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 SLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N, $ IERR ) ELSE CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N, $ IERR ) CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N, $ IERR ) END IF END IF * WORK( 1 ) = MAXWRK RETURN * * End of SGEEVX * END |