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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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| subroutine shseqr | ( | character | job, |
| character | compz, | ||
| integer | n, | ||
| integer | ilo, | ||
| integer | ihi, | ||
| real, dimension( ldh, * ) | h, | ||
| integer | ldh, | ||
| real, dimension( * ) | wr, | ||
| real, dimension( * ) | wi, | ||
| real, dimension( ldz, * ) | z, | ||
| integer | ldz, | ||
| real, dimension( * ) | work, | ||
| integer | lwork, | ||
| integer | info ) |
SHSEQR
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!> !> 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. !>
| [in] | JOB | !> JOB is CHARACTER*1 !> = 'E': compute eigenvalues only; !> = 'S': compute eigenvalues and the Schur form T. !> |
| [in] | COMPZ | !> COMPZ is 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. !> |
| [in] | N | !> N is INTEGER !> The order of the matrix H. N >= 0. !> |
| [in] | ILO | !> ILO is INTEGER !> |
| [in] | IHI | !> IHI is 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 ZGEHRD !> 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 > 0, then 1 <= ILO <= IHI <= N. !> If N = 0, then ILO = 1 and IHI = 0. !> |
| [in,out] | H | !> H is 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) < 0. If INFO = 0 and JOB = 'E', the !> contents of H are unspecified on exit. (The output value of !> H when INFO > 0 is given under the description of INFO !> below.) !> !> Unlike earlier versions of SHSEQR, this subroutine may !> explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1 !> or j = IHI+1, IHI+2, ... N. !> |
| [in] | LDH | !> LDH is INTEGER !> The leading dimension of the array H. LDH >= max(1,N). !> |
| [out] | WR | !> WR is REAL array, dimension (N) !> |
| [out] | WI | !> WI is 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) > 0 and !> WI(i+1) < 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). !> |
| [in,out] | Z | !> Z is 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 > 0 is given under !> the description of INFO below.) !> |
| [in] | LDZ | !> LDZ is INTEGER !> The leading dimension of the array Z. if COMPZ = 'I' or !> COMPZ = 'V', then LDZ >= MAX(1,N). Otherwise, LDZ >= 1. !> |
| [out] | WORK | !> WORK is REAL array, dimension (LWORK) !> On exit, if INFO = 0, WORK(1) returns an estimate of !> the optimal value for LWORK. !> |
| [in] | LWORK | !> LWORK is INTEGER !> The dimension of the array WORK. LWORK >= 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. !> |
| [out] | INFO | !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal !> value !> > 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 > 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 > 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 > 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 > 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 > 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) <= 500 is NS. !> The default for (IHI-ILO+1) > 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. !>
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.
Definition at line 312 of file shseqr.f.
1.11.0