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---
:name: dlaqr4
:md5sum: d8ba6b6e3c95815f0a98bf31c88ffd41
:category: :subroutine
:arguments:
- wantt:
:type: logical
:intent: input
- wantz:
:type: logical
:intent: input
- n:
:type: integer
:intent: input
- ilo:
:type: integer
:intent: input
- ihi:
:type: integer
:intent: input
- h:
:type: doublereal
:intent: input/output
:dims:
- ldh
- n
- ldh:
:type: integer
:intent: input
- wr:
:type: doublereal
:intent: output
:dims:
- ihi
- wi:
:type: doublereal
:intent: output
:dims:
- ihi
- iloz:
:type: integer
:intent: input
- ihiz:
:type: integer
:intent: input
- z:
:type: doublereal
:intent: input/output
:dims:
- ldz
- ihi
- ldz:
:type: integer
:intent: input
- work:
:type: doublereal
:intent: output
:dims:
- MAX(1,lwork)
- lwork:
:type: integer
:intent: input
:option: true
:default: n
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* DLAQR4 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**T, where T is an upper quasi-triangular matrix (the\n\
* Schur form), and Z is the orthogonal matrix of Schur vectors.\n\
*\n\
* Optionally Z may be postmultiplied into an input orthogonal\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 orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* WANTT (input) LOGICAL\n\
* = .TRUE. : the full Schur form T is required;\n\
* = .FALSE.: only eigenvalues are required.\n\
*\n\
* WANTZ (input) LOGICAL\n\
* = .TRUE. : the matrix of Schur vectors Z is required;\n\
* = .FALSE.: Schur vectors are not required.\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 and, if ILO.GT.1,\n\
* H(ILO,ILO-1) is zero. ILO and IHI are normally set by a\n\
* previous call to DGEBAL, and then passed to DGEHRD when the\n\
* matrix output by DGEBAL is reduced to Hessenberg form.\n\
* 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) DOUBLE PRECISION array, dimension (LDH,N)\n\
* On entry, the upper Hessenberg matrix H.\n\
* On exit, if INFO = 0 and WANTT is .TRUE., then H contains\n\
* the upper quasi-triangular matrix T from the Schur\n\
* decomposition (the Schur form); 2-by-2 diagonal blocks\n\
* (corresponding to complex conjugate pairs of eigenvalues)\n\
* are returned in standard form, with H(i,i) = H(i+1,i+1)\n\
* and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is\n\
* .FALSE., then the contents of H are unspecified on exit.\n\
* (The output value of H when INFO.GT.0 is given under the\n\
* description of INFO below.)\n\
*\n\
* This subroutine may explicitly set H(i,j) = 0 for i.GT.j and\n\
* j = 1, 2, ... ILO-1 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\
* WR (output) DOUBLE PRECISION array, dimension (IHI)\n\
* WI (output) DOUBLE PRECISION array, dimension (IHI)\n\
* The real and imaginary parts, respectively, of the computed\n\
* eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)\n\
* and WI(ILO:IHI). If two eigenvalues are computed as a\n\
* complex conjugate pair, they are stored in consecutive\n\
* elements of WR and WI, say the i-th and (i+1)th, with\n\
* WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then\n\
* the eigenvalues are stored in the same order as on the\n\
* diagonal of the Schur form returned in H, with\n\
* WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal\n\
* block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and\n\
* WI(i+1) = -WI(i).\n\
*\n\
* ILOZ (input) INTEGER\n\
* IHIZ (input) INTEGER\n\
* Specify the rows of Z to which transformations must be\n\
* applied if WANTZ is .TRUE..\n\
* 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.\n\
*\n\
* Z (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI)\n\
* If WANTZ is .FALSE., then Z is not referenced.\n\
* If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is\n\
* replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the\n\
* orthogonal Schur factor of H(ILO:IHI,ILO:IHI).\n\
* (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 WANTZ is .TRUE.\n\
* then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.\n\
*\n\
* WORK (workspace/output) DOUBLE PRECISION array, dimension LWORK\n\
* On exit, if LWORK = -1, 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, but LWORK typically as large as 6*N may\n\
* be required for optimal performance. A workspace query\n\
* to determine the optimal workspace size is recommended.\n\
*\n\
* If LWORK = -1, then DLAQR4 does a workspace query.\n\
* In this case, DLAQR4 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\
* .GT. 0: if INFO = i, DLAQR4 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 WANT is .FALSE., then on exit,\n\
* the 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 WANTT is .TRUE., then on exit\n\
*\n\
* (*) (initial value of H)*U = U*(final value of H)\n\
*\n\
* where U is an orthogonal matrix. The final\n\
* value of H is upper Hessenberg and quasi-triangular\n\
* in rows and columns INFO+1 through IHI.\n\
*\n\
* If INFO .GT. 0 and WANTZ is .TRUE., then on exit\n\
*\n\
* (final value of Z(ILO:IHI,ILOZ:IHIZ)\n\
* = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U\n\
*\n\
* where U is the orthogonal matrix in (*) (regard-\n\
* less of the value of WANTT.)\n\
*\n\
* If INFO .GT. 0 and WANTZ is .FALSE., then Z is not\n\
* accessed.\n\
*\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"
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