SUBROUTINE SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, \$ LDZ, WORK, LWORK, INFO ) * * -- LAPACK computational routine (version 3.2.2) -- * Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.. * June 2010 * * .. Scalar Arguments .. INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N CHARACTER COMPZ, JOB * .. * .. Array Arguments .. REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ), \$ Z( LDZ, * ) * .. * Purpose * ======= * * SHSEQR computes the eigenvalues of a Hessenberg matrix H * and, optionally, the matrices T and Z from the Schur decomposition * H = Z T Z**T, where T is an upper quasi-triangular matrix (the * Schur form), and Z is the orthogonal matrix of Schur vectors. * * Optionally Z may be postmultiplied into an input orthogonal * matrix Q so that this routine can give the Schur factorization * of a matrix A which has been reduced to the Hessenberg form H * by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. * * Arguments * ========= * * JOB (input) CHARACTER*1 * = 'E': compute eigenvalues only; * = 'S': compute eigenvalues and the Schur form T. * * COMPZ (input) CHARACTER*1 * = 'N': no Schur vectors are computed; * = 'I': Z is initialized to the unit matrix and the matrix Z * of Schur vectors of H is returned; * = 'V': Z must contain an orthogonal matrix Q on entry, and * the product Q*Z is returned. * * N (input) INTEGER * The order of the matrix H. N .GE. 0. * * ILO (input) INTEGER * IHI (input) INTEGER * It is assumed that H is already upper triangular in rows * and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally * set by a previous call to SGEBAL, and then passed to SGEHRD * when the matrix output by SGEBAL is reduced to Hessenberg * form. Otherwise ILO and IHI should be set to 1 and N * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. * If N = 0, then ILO = 1 and IHI = 0. * * H (input/output) REAL array, dimension (LDH,N) * On entry, the upper Hessenberg matrix H. * On exit, if INFO = 0 and JOB = 'S', then H contains the * upper quasi-triangular matrix T from the Schur decomposition * (the Schur form); 2-by-2 diagonal blocks (corresponding to * complex conjugate pairs of eigenvalues) are returned in * standard form, with H(i,i) = H(i+1,i+1) and * H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the * contents of H are unspecified on exit. (The output value of * H when INFO.GT.0 is given under the description of INFO * below.) * * Unlike earlier versions of SHSEQR, this subroutine may * explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 * or j = IHI+1, IHI+2, ... N. * * LDH (input) INTEGER * The leading dimension of the array H. LDH .GE. max(1,N). * * WR (output) REAL array, dimension (N) * WI (output) REAL array, dimension (N) * The real and imaginary parts, respectively, of the computed * eigenvalues. If two eigenvalues are computed as a complex * conjugate pair, they are stored in consecutive elements of * WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and * WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in * the same order as on the diagonal of the Schur form returned * in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 * diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and * WI(i+1) = -WI(i). * * Z (input/output) REAL array, dimension (LDZ,N) * If COMPZ = 'N', Z is not referenced. * If COMPZ = 'I', on entry Z need not be set and on exit, * if INFO = 0, Z contains the orthogonal matrix Z of the Schur * vectors of H. If COMPZ = 'V', on entry Z must contain an * N-by-N matrix Q, which is assumed to be equal to the unit * matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, * if INFO = 0, Z contains Q*Z. * Normally Q is the orthogonal matrix generated by SORGHR * after the call to SGEHRD which formed the Hessenberg matrix * H. (The output value of Z when INFO.GT.0 is given under * the description of INFO below.) * * LDZ (input) INTEGER * The leading dimension of the array Z. if COMPZ = 'I' or * COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1. * * WORK (workspace/output) REAL array, dimension (LWORK) * On exit, if INFO = 0, WORK(1) returns an estimate of * the optimal value for LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK .GE. max(1,N) * is sufficient and delivers very good and sometimes * optimal performance. However, LWORK as large as 11*N * may be required for optimal performance. A workspace * query is recommended to determine the optimal workspace * size. * * If LWORK = -1, then SHSEQR does a workspace query. * In this case, SHSEQR checks the input parameters and * estimates the optimal workspace size for the given * values of N, ILO and IHI. The estimate is returned * in WORK(1). No error message related to LWORK is * issued by XERBLA. Neither H nor Z are accessed. * * * INFO (output) INTEGER * = 0: successful exit * .LT. 0: if INFO = -i, the i-th argument had an illegal * value * .GT. 0: if INFO = i, SHSEQR failed to compute all of * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR * and WI contain those eigenvalues which have been * successfully computed. (Failures are rare.) * * If INFO .GT. 0 and JOB = 'E', then on exit, the * remaining unconverged eigenvalues are the eigen- * values of the upper Hessenberg matrix rows and * columns ILO through INFO of the final, output * value of H. * * If INFO .GT. 0 and JOB = 'S', then on exit * * (*) (initial value of H)*U = U*(final value of H) * * where U is an orthogonal matrix. The final * value of H is upper Hessenberg and quasi-triangular * in rows and columns INFO+1 through IHI. * * If INFO .GT. 0 and COMPZ = 'V', then on exit * * (final value of Z) = (initial value of Z)*U * * where U is the orthogonal matrix in (*) (regard- * less of the value of JOB.) * * If INFO .GT. 0 and COMPZ = 'I', then on exit * (final value of Z) = U * where U is the orthogonal matrix in (*) (regard- * less of the value of JOB.) * * If INFO .GT. 0 and COMPZ = 'N', then Z is not * accessed. * * ================================================================ * Default values supplied by * ILAENV(ISPEC,'SHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK). * It is suggested that these defaults be adjusted in order * to attain best performance in each particular * computational environment. * * ISPEC=12: The SLAHQR vs SLAQR0 crossover point. * Default: 75. (Must be at least 11.) * * ISPEC=13: Recommended deflation window size. * This depends on ILO, IHI and NS. NS is the * number of simultaneous shifts returned * by ILAENV(ISPEC=15). (See ISPEC=15 below.) * The default for (IHI-ILO+1).LE.500 is NS. * The default for (IHI-ILO+1).GT.500 is 3*NS/2. * * ISPEC=14: Nibble crossover point. (See IPARMQ for * details.) Default: 14% of deflation window * size. * * ISPEC=15: Number of simultaneous shifts in a multishift * QR iteration. * * If IHI-ILO+1 is ... * * greater than ...but less ... the * or equal to ... than default is * * 1 30 NS = 2(+) * 30 60 NS = 4(+) * 60 150 NS = 10(+) * 150 590 NS = ** * 590 3000 NS = 64 * 3000 6000 NS = 128 * 6000 infinity NS = 256 * * (+) By default some or all matrices of this order * are passed to the implicit double shift routine * SLAHQR and this parameter is ignored. See * ISPEC=12 above and comments in IPARMQ for * details. * * (**) The asterisks (**) indicate an ad-hoc * function of N increasing from 10 to 64. * * ISPEC=16: Select structured matrix multiply. * If the number of simultaneous shifts (specified * by ISPEC=15) is less than 14, then the default * for ISPEC=16 is 0. Otherwise the default for * ISPEC=16 is 2. * * ================================================================ * Based on contributions by * Karen Braman and Ralph Byers, Department of Mathematics, * University of Kansas, USA * * ================================================================ * References: * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR * Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 * Performance, SIAM Journal of Matrix Analysis, volume 23, pages * 929--947, 2002. * * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR * Algorithm Part II: Aggressive Early Deflation, SIAM Journal * of Matrix Analysis, volume 23, pages 948--973, 2002. * * ================================================================ * .. Parameters .. * * ==== Matrices of order NTINY or smaller must be processed by * . SLAHQR because of insufficient subdiagonal scratch space. * . (This is a hard limit.) ==== INTEGER NTINY PARAMETER ( NTINY = 11 ) * * ==== NL allocates some local workspace to help small matrices * . through a rare SLAHQR failure. NL .GT. NTINY = 11 is * . required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom- * . mended. (The default value of NMIN is 75.) Using NL = 49 * . allows up to six simultaneous shifts and a 16-by-16 * . deflation window. ==== INTEGER NL PARAMETER ( NL = 49 ) REAL ZERO, ONE PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0 ) * .. * .. Local Arrays .. REAL HL( NL, NL ), WORKL( NL ) * .. * .. Local Scalars .. INTEGER I, KBOT, NMIN LOGICAL INITZ, LQUERY, WANTT, WANTZ * .. * .. External Functions .. INTEGER ILAENV LOGICAL LSAME EXTERNAL ILAENV, LSAME * .. * .. External Subroutines .. EXTERNAL SLACPY, SLAHQR, SLAQR0, SLASET, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, REAL * .. * .. Executable Statements .. * * ==== Decode and check the input parameters. ==== * WANTT = LSAME( JOB, 'S' ) INITZ = LSAME( COMPZ, 'I' ) WANTZ = INITZ .OR. LSAME( COMPZ, 'V' ) WORK( 1 ) = REAL( MAX( 1, N ) ) LQUERY = LWORK.EQ.-1 * INFO = 0 IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN INFO = -1 ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -5 ELSE IF( LDH.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN INFO = -11 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -13 END IF * IF( INFO.NE.0 ) THEN * * ==== Quick return in case of invalid argument. ==== * CALL XERBLA( 'SHSEQR', -INFO ) RETURN * ELSE IF( N.EQ.0 ) THEN * * ==== Quick return in case N = 0; nothing to do. ==== * RETURN * ELSE IF( LQUERY ) THEN * * ==== Quick return in case of a workspace query ==== * CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO, \$ IHI, Z, LDZ, WORK, LWORK, INFO ) * ==== Ensure reported workspace size is backward-compatible with * . previous LAPACK versions. ==== WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) ) RETURN * ELSE * * ==== copy eigenvalues isolated by SGEBAL ==== * DO 10 I = 1, ILO - 1 WR( I ) = H( I, I ) WI( I ) = ZERO 10 CONTINUE DO 20 I = IHI + 1, N WR( I ) = H( I, I ) WI( I ) = ZERO 20 CONTINUE * * ==== Initialize Z, if requested ==== * IF( INITZ ) \$ CALL SLASET( 'A', N, N, ZERO, ONE, Z, LDZ ) * * ==== Quick return if possible ==== * IF( ILO.EQ.IHI ) THEN WR( ILO ) = H( ILO, ILO ) WI( ILO ) = ZERO RETURN END IF * * ==== SLAHQR/SLAQR0 crossover point ==== * NMIN = ILAENV( 12, 'SHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N, \$ ILO, IHI, LWORK ) NMIN = MAX( NTINY, NMIN ) * * ==== SLAQR0 for big matrices; SLAHQR for small ones ==== * IF( N.GT.NMIN ) THEN CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO, \$ IHI, Z, LDZ, WORK, LWORK, INFO ) ELSE * * ==== Small matrix ==== * CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO, \$ IHI, Z, LDZ, INFO ) * IF( INFO.GT.0 ) THEN * * ==== A rare SLAHQR failure! SLAQR0 sometimes succeeds * . when SLAHQR fails. ==== * KBOT = INFO * IF( N.GE.NL ) THEN * * ==== Larger matrices have enough subdiagonal scratch * . space to call SLAQR0 directly. ==== * CALL SLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, WR, \$ WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO ) * ELSE * * ==== Tiny matrices don't have enough subdiagonal * . scratch space to benefit from SLAQR0. Hence, * . tiny matrices must be copied into a larger * . array before calling SLAQR0. ==== * CALL SLACPY( 'A', N, N, H, LDH, HL, NL ) HL( N+1, N ) = ZERO CALL SLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ), \$ NL ) CALL SLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, WR, \$ WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO ) IF( WANTT .OR. INFO.NE.0 ) \$ CALL SLACPY( 'A', N, N, HL, NL, H, LDH ) END IF END IF END IF * * ==== Clear out the trash, if necessary. ==== * IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 ) \$ CALL SLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH ) * * ==== Ensure reported workspace size is backward-compatible with * . previous LAPACK versions. ==== * WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) ) END IF * * ==== End of SHSEQR ==== * END