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---
:name: zhseqr
:md5sum: ef6b85e6bdbf0fa44c0c2503d7c14f6f
:category: :subroutine
:arguments:
- job:
:type: char
:intent: input
- compz:
:type: char
:intent: input
- n:
:type: integer
:intent: input
- ilo:
:type: integer
:intent: input
- ihi:
:type: integer
:intent: input
- h:
:type: doublecomplex
:intent: input/output
:dims:
- ldh
- n
- ldh:
:type: integer
:intent: input
- w:
:type: doublecomplex
:intent: output
:dims:
- n
- z:
:type: doublecomplex
:intent: input/output
:dims:
- "lsame_(&compz,\"N\") ? 0 : ldz"
- "lsame_(&compz,\"N\") ? 0 : n"
- ldz:
:type: integer
:intent: input
- work:
:type: doublecomplex
:intent: output
:dims:
- MAX(1,lwork)
- lwork:
:type: integer
:intent: input
:option: true
:default: n
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* ZHSEQR computes the eigenvalues of a Hessenberg matrix H\n\
* and, optionally, the matrices T and Z from the Schur decomposition\n\
* H = Z T Z**H, where T is an upper triangular matrix (the\n\
* Schur form), and Z is the unitary matrix of Schur vectors.\n\
*\n\
* Optionally Z may be postmultiplied into an input unitary\n\
* matrix Q so that this routine can give the Schur factorization\n\
* of a matrix A which has been reduced to the Hessenberg form H\n\
* by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* JOB (input) CHARACTER*1\n\
* = 'E': compute eigenvalues only;\n\
* = 'S': compute eigenvalues and the Schur form T.\n\
*\n\
* COMPZ (input) CHARACTER*1\n\
* = 'N': no Schur vectors are computed;\n\
* = 'I': Z is initialized to the unit matrix and the matrix Z\n\
* of Schur vectors of H is returned;\n\
* = 'V': Z must contain an unitary matrix Q on entry, and\n\
* the product Q*Z is returned.\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix H. N .GE. 0.\n\
*\n\
* ILO (input) INTEGER\n\
* IHI (input) INTEGER\n\
* It is assumed that H is already upper triangular in rows\n\
* and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally\n\
* set by a previous call to ZGEBAL, and then passed to ZGEHRD\n\
* when the matrix output by ZGEBAL is reduced to Hessenberg\n\
* form. Otherwise ILO and IHI should be set to 1 and N\n\
* respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.\n\
* If N = 0, then ILO = 1 and IHI = 0.\n\
*\n\
* H (input/output) COMPLEX*16 array, dimension (LDH,N)\n\
* On entry, the upper Hessenberg matrix H.\n\
* On exit, if INFO = 0 and JOB = 'S', H contains the upper\n\
* triangular matrix T from the Schur decomposition (the\n\
* Schur form). If INFO = 0 and JOB = 'E', the contents of\n\
* H are unspecified on exit. (The output value of H when\n\
* INFO.GT.0 is given under the description of INFO below.)\n\
*\n\
* Unlike earlier versions of ZHSEQR, this subroutine may\n\
* explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1\n\
* or j = IHI+1, IHI+2, ... N.\n\
*\n\
* LDH (input) INTEGER\n\
* The leading dimension of the array H. LDH .GE. max(1,N).\n\
*\n\
* W (output) COMPLEX*16 array, dimension (N)\n\
* The computed eigenvalues. If JOB = 'S', the eigenvalues are\n\
* stored in the same order as on the diagonal of the Schur\n\
* form returned in H, with W(i) = H(i,i).\n\
*\n\
* Z (input/output) COMPLEX*16 array, dimension (LDZ,N)\n\
* If COMPZ = 'N', Z is not referenced.\n\
* If COMPZ = 'I', on entry Z need not be set and on exit,\n\
* if INFO = 0, Z contains the unitary matrix Z of the Schur\n\
* vectors of H. If COMPZ = 'V', on entry Z must contain an\n\
* N-by-N matrix Q, which is assumed to be equal to the unit\n\
* matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,\n\
* if INFO = 0, Z contains Q*Z.\n\
* Normally Q is the unitary matrix generated by ZUNGHR\n\
* after the call to ZGEHRD which formed the Hessenberg matrix\n\
* H. (The output value of Z when INFO.GT.0 is given under\n\
* the description of INFO below.)\n\
*\n\
* LDZ (input) INTEGER\n\
* The leading dimension of the array Z. if COMPZ = 'I' or\n\
* COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.\n\
*\n\
* WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)\n\
* On exit, if INFO = 0, WORK(1) returns an estimate of\n\
* the optimal value for LWORK.\n\
*\n\
* LWORK (input) INTEGER\n\
* The dimension of the array WORK. LWORK .GE. max(1,N)\n\
* is sufficient and delivers very good and sometimes\n\
* optimal performance. However, LWORK as large as 11*N\n\
* may be required for optimal performance. A workspace\n\
* query is recommended to determine the optimal workspace\n\
* size.\n\
*\n\
* If LWORK = -1, then ZHSEQR does a workspace query.\n\
* In this case, ZHSEQR checks the input parameters and\n\
* estimates the optimal workspace size for the given\n\
* values of N, ILO and IHI. The estimate is returned\n\
* in WORK(1). No error message related to LWORK is\n\
* issued by XERBLA. Neither H nor Z are accessed.\n\
*\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit\n\
* .LT. 0: if INFO = -i, the i-th argument had an illegal\n\
* value\n\
* .GT. 0: if INFO = i, ZHSEQR failed to compute all of\n\
* the eigenvalues. Elements 1:ilo-1 and i+1:n of WR\n\
* and WI contain those eigenvalues which have been\n\
* successfully computed. (Failures are rare.)\n\
*\n\
* If INFO .GT. 0 and JOB = 'E', then on exit, the\n\
* remaining unconverged eigenvalues are the eigen-\n\
* values of the upper Hessenberg matrix rows and\n\
* columns ILO through INFO of the final, output\n\
* value of H.\n\
*\n\
* If INFO .GT. 0 and JOB = 'S', then on exit\n\
*\n\
* (*) (initial value of H)*U = U*(final value of H)\n\
*\n\
* where U is a unitary matrix. The final\n\
* value of H is upper Hessenberg and triangular in\n\
* rows and columns INFO+1 through IHI.\n\
*\n\
* If INFO .GT. 0 and COMPZ = 'V', then on exit\n\
*\n\
* (final value of Z) = (initial value of Z)*U\n\
*\n\
* where U is the unitary matrix in (*) (regard-\n\
* less of the value of JOB.)\n\
*\n\
* If INFO .GT. 0 and COMPZ = 'I', then on exit\n\
* (final value of Z) = U\n\
* where U is the unitary matrix in (*) (regard-\n\
* less of the value of JOB.)\n\
*\n\
* If INFO .GT. 0 and COMPZ = 'N', then Z is not\n\
* accessed.\n\
*\n\n\
* ================================================================\n\
* Default values supplied by\n\
* ILAENV(ISPEC,'ZHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).\n\
* It is suggested that these defaults be adjusted in order\n\
* to attain best performance in each particular\n\
* computational environment.\n\
*\n\
* ISPEC=12: The ZLAHQR vs ZLAQR0 crossover point.\n\
* Default: 75. (Must be at least 11.)\n\
*\n\
* ISPEC=13: Recommended deflation window size.\n\
* This depends on ILO, IHI and NS. NS is the\n\
* number of simultaneous shifts returned\n\
* by ILAENV(ISPEC=15). (See ISPEC=15 below.)\n\
* The default for (IHI-ILO+1).LE.500 is NS.\n\
* The default for (IHI-ILO+1).GT.500 is 3*NS/2.\n\
*\n\
* ISPEC=14: Nibble crossover point. (See IPARMQ for\n\
* details.) Default: 14% of deflation window\n\
* size.\n\
*\n\
* ISPEC=15: Number of simultaneous shifts in a multishift\n\
* QR iteration.\n\
*\n\
* If IHI-ILO+1 is ...\n\
*\n\
* greater than ...but less ... the\n\
* or equal to ... than default is\n\
*\n\
* 1 30 NS = 2(+)\n\
* 30 60 NS = 4(+)\n\
* 60 150 NS = 10(+)\n\
* 150 590 NS = **\n\
* 590 3000 NS = 64\n\
* 3000 6000 NS = 128\n\
* 6000 infinity NS = 256\n\
*\n\
* (+) By default some or all matrices of this order\n\
* are passed to the implicit double shift routine\n\
* ZLAHQR and this parameter is ignored. See\n\
* ISPEC=12 above and comments in IPARMQ for\n\
* details.\n\
*\n\
* (**) The asterisks (**) indicate an ad-hoc\n\
* function of N increasing from 10 to 64.\n\
*\n\
* ISPEC=16: Select structured matrix multiply.\n\
* If the number of simultaneous shifts (specified\n\
* by ISPEC=15) is less than 14, then the default\n\
* for ISPEC=16 is 0. Otherwise the default for\n\
* ISPEC=16 is 2.\n\
*\n\
* ================================================================\n\
* Based on contributions by\n\
* Karen Braman and Ralph Byers, Department of Mathematics,\n\
* University of Kansas, USA\n\
*\n\
* ================================================================\n\
* References:\n\
* K. Braman, R. Byers and R. Mathias, The Multi-Shift QR\n\
* Algorithm Part I: Maintaining Well Focused Shifts, and Level 3\n\
* Performance, SIAM Journal of Matrix Analysis, volume 23, pages\n\
* 929--947, 2002.\n\
*\n\
* K. Braman, R. Byers and R. Mathias, The Multi-Shift QR\n\
* Algorithm Part II: Aggressive Early Deflation, SIAM Journal\n\
* of Matrix Analysis, volume 23, pages 948--973, 2002.\n\
*\n\
* ================================================================\n"
|
