641 lines
24 KiB
FortranFixed
641 lines
24 KiB
FortranFixed
SUBROUTINE SLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
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$ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
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*
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* -- LAPACK auxiliary routine (version 3.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* November 2006
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*
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* .. Scalar Arguments ..
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INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
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LOGICAL WANTT, WANTZ
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* ..
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* .. Array Arguments ..
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REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ),
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$ Z( LDZ, * )
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* ..
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*
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* This subroutine implements one level of recursion for SLAQR0.
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* It is a complete implementation of the small bulge multi-shift
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* QR algorithm. It may be called by SLAQR0 and, for large enough
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* deflation window size, it may be called by SLAQR3. This
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* subroutine is identical to SLAQR0 except that it calls SLAQR2
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* instead of SLAQR3.
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*
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* Purpose
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* =======
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*
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* SLAQR4 computes the eigenvalues of a Hessenberg matrix H
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* and, optionally, the matrices T and Z from the Schur decomposition
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* H = Z T Z**T, where T is an upper quasi-triangular matrix (the
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* Schur form), and Z is the orthogonal matrix of Schur vectors.
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*
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* Optionally Z may be postmultiplied into an input orthogonal
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* matrix Q so that this routine can give the Schur factorization
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* of a matrix A which has been reduced to the Hessenberg form H
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* by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
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*
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* Arguments
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* =========
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*
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* WANTT (input) LOGICAL
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* = .TRUE. : the full Schur form T is required;
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* = .FALSE.: only eigenvalues are required.
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*
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* WANTZ (input) LOGICAL
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* = .TRUE. : the matrix of Schur vectors Z is required;
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* = .FALSE.: Schur vectors are not required.
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*
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* N (input) INTEGER
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* The order of the matrix H. N .GE. 0.
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*
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* ILO (input) INTEGER
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* IHI (input) INTEGER
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* It is assumed that H is already upper triangular in rows
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* and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
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* H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
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* previous call to SGEBAL, and then passed to SGEHRD when the
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* matrix output by SGEBAL is reduced to Hessenberg form.
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* Otherwise, ILO and IHI should be set to 1 and N,
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* respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
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* If N = 0, then ILO = 1 and IHI = 0.
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*
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* H (input/output) REAL array, dimension (LDH,N)
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* On entry, the upper Hessenberg matrix H.
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* On exit, if INFO = 0 and WANTT is .TRUE., then H contains
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* the upper quasi-triangular matrix T from the Schur
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* decomposition (the Schur form); 2-by-2 diagonal blocks
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* (corresponding to complex conjugate pairs of eigenvalues)
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* are returned in standard form, with H(i,i) = H(i+1,i+1)
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* and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
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* .FALSE., then the contents of H are unspecified on exit.
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* (The output value of H when INFO.GT.0 is given under the
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* description of INFO below.)
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*
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* This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
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* j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
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*
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* LDH (input) INTEGER
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* The leading dimension of the array H. LDH .GE. max(1,N).
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*
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* WR (output) REAL array, dimension (IHI)
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* WI (output) REAL array, dimension (IHI)
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* The real and imaginary parts, respectively, of the computed
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* eigenvalues of H(ILO:IHI,ILO:IHI) are stored WR(ILO:IHI)
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* and WI(ILO:IHI). If two eigenvalues are computed as a
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* complex conjugate pair, they are stored in consecutive
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* elements of WR and WI, say the i-th and (i+1)th, with
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* WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
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* the eigenvalues are stored in the same order as on the
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* diagonal of the Schur form returned in H, with
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* WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
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* block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
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* WI(i+1) = -WI(i).
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*
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* ILOZ (input) INTEGER
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* IHIZ (input) INTEGER
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* Specify the rows of Z to which transformations must be
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* applied if WANTZ is .TRUE..
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* 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
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*
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* Z (input/output) REAL array, dimension (LDZ,IHI)
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* If WANTZ is .FALSE., then Z is not referenced.
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* If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
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* replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
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* orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
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* (The output value of Z when INFO.GT.0 is given under
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* the description of INFO below.)
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*
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* LDZ (input) INTEGER
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* The leading dimension of the array Z. if WANTZ is .TRUE.
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* then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
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*
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* WORK (workspace/output) REAL array, dimension LWORK
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* On exit, if LWORK = -1, WORK(1) returns an estimate of
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* the optimal value for LWORK.
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*
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* LWORK (input) INTEGER
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* The dimension of the array WORK. LWORK .GE. max(1,N)
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* is sufficient, but LWORK typically as large as 6*N may
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* be required for optimal performance. A workspace query
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* to determine the optimal workspace size is recommended.
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*
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* If LWORK = -1, then SLAQR4 does a workspace query.
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* In this case, SLAQR4 checks the input parameters and
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* estimates the optimal workspace size for the given
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* values of N, ILO and IHI. The estimate is returned
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* in WORK(1). No error message related to LWORK is
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* issued by XERBLA. Neither H nor Z are accessed.
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*
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*
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* INFO (output) INTEGER
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* = 0: successful exit
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* .GT. 0: if INFO = i, SLAQR4 failed to compute all of
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* the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
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* and WI contain those eigenvalues which have been
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* successfully computed. (Failures are rare.)
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*
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* If INFO .GT. 0 and WANT is .FALSE., then on exit,
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* the remaining unconverged eigenvalues are the eigen-
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* values of the upper Hessenberg matrix rows and
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* columns ILO through INFO of the final, output
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* value of H.
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*
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* If INFO .GT. 0 and WANTT is .TRUE., then on exit
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*
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* (*) (initial value of H)*U = U*(final value of H)
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*
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* where U is an orthogonal matrix. The final
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* value of H is upper Hessenberg and quasi-triangular
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* in rows and columns INFO+1 through IHI.
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*
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* If INFO .GT. 0 and WANTZ is .TRUE., then on exit
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*
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* (final value of Z(ILO:IHI,ILOZ:IHIZ)
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* = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
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*
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* where U is the orthogonal matrix in (*) (regard-
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* less of the value of WANTT.)
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*
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* If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
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* accessed.
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*
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* ================================================================
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* Based on contributions by
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* Karen Braman and Ralph Byers, Department of Mathematics,
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* University of Kansas, USA
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*
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* ================================================================
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* References:
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* K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
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* Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
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* Performance, SIAM Journal of Matrix Analysis, volume 23, pages
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* 929--947, 2002.
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*
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* K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
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* Algorithm Part II: Aggressive Early Deflation, SIAM Journal
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* of Matrix Analysis, volume 23, pages 948--973, 2002.
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*
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* ================================================================
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* .. Parameters ..
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*
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* ==== Matrices of order NTINY or smaller must be processed by
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* . SLAHQR because of insufficient subdiagonal scratch space.
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* . (This is a hard limit.) ====
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*
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* ==== Exceptional deflation windows: try to cure rare
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* . slow convergence by increasing the size of the
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* . deflation window after KEXNW iterations. =====
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*
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* ==== Exceptional shifts: try to cure rare slow convergence
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* . with ad-hoc exceptional shifts every KEXSH iterations.
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* . The constants WILK1 and WILK2 are used to form the
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* . exceptional shifts. ====
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*
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INTEGER NTINY
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PARAMETER ( NTINY = 11 )
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INTEGER KEXNW, KEXSH
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PARAMETER ( KEXNW = 5, KEXSH = 6 )
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REAL WILK1, WILK2
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PARAMETER ( WILK1 = 0.75e0, WILK2 = -0.4375e0 )
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0 )
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* ..
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* .. Local Scalars ..
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REAL AA, BB, CC, CS, DD, SN, SS, SWAP
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INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
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$ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
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$ LWKOPT, NDFL, NH, NHO, NIBBLE, NMIN, NS, NSMAX,
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$ NSR, NVE, NW, NWMAX, NWR
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LOGICAL NWINC, SORTED
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CHARACTER JBCMPZ*2
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* ..
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* .. External Functions ..
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INTEGER ILAENV
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EXTERNAL ILAENV
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* ..
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* .. Local Arrays ..
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REAL ZDUM( 1, 1 )
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* ..
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* .. External Subroutines ..
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EXTERNAL SLACPY, SLAHQR, SLANV2, SLAQR2, SLAQR5
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, INT, MAX, MIN, MOD, REAL
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* ..
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* .. Executable Statements ..
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INFO = 0
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*
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* ==== Quick return for N = 0: nothing to do. ====
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*
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IF( N.EQ.0 ) THEN
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WORK( 1 ) = ONE
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RETURN
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END IF
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*
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* ==== Set up job flags for ILAENV. ====
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*
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IF( WANTT ) THEN
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JBCMPZ( 1: 1 ) = 'S'
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ELSE
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JBCMPZ( 1: 1 ) = 'E'
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END IF
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IF( WANTZ ) THEN
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JBCMPZ( 2: 2 ) = 'V'
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ELSE
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JBCMPZ( 2: 2 ) = 'N'
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END IF
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*
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* ==== Tiny matrices must use SLAHQR. ====
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*
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IF( N.LE.NTINY ) THEN
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*
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* ==== Estimate optimal workspace. ====
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*
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LWKOPT = 1
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IF( LWORK.NE.-1 )
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$ CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
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$ ILOZ, IHIZ, Z, LDZ, INFO )
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ELSE
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*
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* ==== Use small bulge multi-shift QR with aggressive early
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* . deflation on larger-than-tiny matrices. ====
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*
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* ==== Hope for the best. ====
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*
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INFO = 0
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*
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* ==== NWR = recommended deflation window size. At this
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* . point, N .GT. NTINY = 11, so there is enough
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* . subdiagonal workspace for NWR.GE.2 as required.
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* . (In fact, there is enough subdiagonal space for
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* . NWR.GE.3.) ====
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*
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NWR = ILAENV( 13, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
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NWR = MAX( 2, NWR )
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NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
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NW = NWR
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*
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* ==== NSR = recommended number of simultaneous shifts.
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* . At this point N .GT. NTINY = 11, so there is at
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* . enough subdiagonal workspace for NSR to be even
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* . and greater than or equal to two as required. ====
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*
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NSR = ILAENV( 15, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
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NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
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NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
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*
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* ==== Estimate optimal workspace ====
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*
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* ==== Workspace query call to SLAQR2 ====
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*
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CALL SLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
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$ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
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$ N, H, LDH, WORK, -1 )
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*
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* ==== Optimal workspace = MAX(SLAQR5, SLAQR2) ====
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*
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LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
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*
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* ==== Quick return in case of workspace query. ====
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*
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IF( LWORK.EQ.-1 ) THEN
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WORK( 1 ) = REAL( LWKOPT )
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RETURN
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END IF
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*
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* ==== SLAHQR/SLAQR0 crossover point ====
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*
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NMIN = ILAENV( 12, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
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NMIN = MAX( NTINY, NMIN )
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*
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* ==== Nibble crossover point ====
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*
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NIBBLE = ILAENV( 14, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
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NIBBLE = MAX( 0, NIBBLE )
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*
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* ==== Accumulate reflections during ttswp? Use block
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* . 2-by-2 structure during matrix-matrix multiply? ====
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*
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KACC22 = ILAENV( 16, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
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KACC22 = MAX( 0, KACC22 )
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KACC22 = MIN( 2, KACC22 )
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*
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* ==== NWMAX = the largest possible deflation window for
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* . which there is sufficient workspace. ====
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*
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NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
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*
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* ==== NSMAX = the Largest number of simultaneous shifts
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* . for which there is sufficient workspace. ====
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*
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NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
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NSMAX = NSMAX - MOD( NSMAX, 2 )
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*
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* ==== NDFL: an iteration count restarted at deflation. ====
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*
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NDFL = 1
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*
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* ==== ITMAX = iteration limit ====
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*
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ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
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*
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* ==== Last row and column in the active block ====
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*
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KBOT = IHI
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*
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* ==== Main Loop ====
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*
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DO 80 IT = 1, ITMAX
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*
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* ==== Done when KBOT falls below ILO ====
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*
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IF( KBOT.LT.ILO )
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$ GO TO 90
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*
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* ==== Locate active block ====
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*
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DO 10 K = KBOT, ILO + 1, -1
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IF( H( K, K-1 ).EQ.ZERO )
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$ GO TO 20
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10 CONTINUE
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K = ILO
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20 CONTINUE
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KTOP = K
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*
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* ==== Select deflation window size ====
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*
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NH = KBOT - KTOP + 1
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IF( NDFL.LT.KEXNW .OR. NH.LT.NW ) THEN
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*
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* ==== Typical deflation window. If possible and
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* . advisable, nibble the entire active block.
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* . If not, use size NWR or NWR+1 depending upon
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* . which has the smaller corresponding subdiagonal
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* . entry (a heuristic). ====
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*
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NWINC = .TRUE.
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IF( NH.LE.MIN( NMIN, NWMAX ) ) THEN
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NW = NH
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ELSE
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NW = MIN( NWR, NH, NWMAX )
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IF( NW.LT.NWMAX ) THEN
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IF( NW.GE.NH-1 ) THEN
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NW = NH
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ELSE
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KWTOP = KBOT - NW + 1
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IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
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$ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
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END IF
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END IF
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END IF
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ELSE
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*
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* ==== Exceptional deflation window. If there have
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* . been no deflations in KEXNW or more iterations,
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* . then vary the deflation window size. At first,
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* . because, larger windows are, in general, more
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* . powerful than smaller ones, rapidly increase the
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* . window up to the maximum reasonable and possible.
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* . Then maybe try a slightly smaller window. ====
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*
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IF( NWINC .AND. NW.LT.MIN( NWMAX, NH ) ) THEN
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NW = MIN( NWMAX, NH, 2*NW )
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ELSE
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NWINC = .FALSE.
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IF( NW.EQ.NH .AND. NH.GT.2 )
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$ NW = NH - 1
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END IF
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END IF
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*
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* ==== Aggressive early deflation:
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* . split workspace under the subdiagonal into
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* . - an nw-by-nw work array V in the lower
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* . left-hand-corner,
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* . - an NW-by-at-least-NW-but-more-is-better
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* . (NW-by-NHO) horizontal work array along
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* . the bottom edge,
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* . - an at-least-NW-but-more-is-better (NHV-by-NW)
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* . vertical work array along the left-hand-edge.
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* . ====
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*
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KV = N - NW + 1
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KT = NW + 1
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NHO = ( N-NW-1 ) - KT + 1
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KWV = NW + 2
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NVE = ( N-NW ) - KWV + 1
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*
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* ==== Aggressive early deflation ====
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*
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CALL SLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
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$ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
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$ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
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$ WORK, LWORK )
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*
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* ==== Adjust KBOT accounting for new deflations. ====
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*
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KBOT = KBOT - LD
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*
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* ==== KS points to the shifts. ====
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*
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KS = KBOT - LS + 1
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*
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* ==== Skip an expensive QR sweep if there is a (partly
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* . heuristic) reason to expect that many eigenvalues
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* . will deflate without it. Here, the QR sweep is
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* . skipped if many eigenvalues have just been deflated
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* . or if the remaining active block is small.
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*
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IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
|
|
$ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
|
|
*
|
|
* ==== NS = nominal number of simultaneous shifts.
|
|
* . This may be lowered (slightly) if SLAQR2
|
|
* . did not provide that many shifts. ====
|
|
*
|
|
NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
|
|
NS = NS - MOD( NS, 2 )
|
|
*
|
|
* ==== If there have been no deflations
|
|
* . in a multiple of KEXSH iterations,
|
|
* . then try exceptional shifts.
|
|
* . Otherwise use shifts provided by
|
|
* . SLAQR2 above or from the eigenvalues
|
|
* . of a trailing principal submatrix. ====
|
|
*
|
|
IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
|
|
KS = KBOT - NS + 1
|
|
DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
|
|
SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
|
|
AA = WILK1*SS + H( I, I )
|
|
BB = SS
|
|
CC = WILK2*SS
|
|
DD = AA
|
|
CALL SLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
|
|
$ WR( I ), WI( I ), CS, SN )
|
|
30 CONTINUE
|
|
IF( KS.EQ.KTOP ) THEN
|
|
WR( KS+1 ) = H( KS+1, KS+1 )
|
|
WI( KS+1 ) = ZERO
|
|
WR( KS ) = WR( KS+1 )
|
|
WI( KS ) = WI( KS+1 )
|
|
END IF
|
|
ELSE
|
|
*
|
|
* ==== Got NS/2 or fewer shifts? Use SLAHQR
|
|
* . on a trailing principal submatrix to
|
|
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
|
|
* . there is enough space below the subdiagonal
|
|
* . to fit an NS-by-NS scratch array.) ====
|
|
*
|
|
IF( KBOT-KS+1.LE.NS / 2 ) THEN
|
|
KS = KBOT - NS + 1
|
|
KT = N - NS + 1
|
|
CALL SLACPY( 'A', NS, NS, H( KS, KS ), LDH,
|
|
$ H( KT, 1 ), LDH )
|
|
CALL SLAHQR( .false., .false., NS, 1, NS,
|
|
$ H( KT, 1 ), LDH, WR( KS ), WI( KS ),
|
|
$ 1, 1, ZDUM, 1, INF )
|
|
KS = KS + INF
|
|
*
|
|
* ==== In case of a rare QR failure use
|
|
* . eigenvalues of the trailing 2-by-2
|
|
* . principal submatrix. ====
|
|
*
|
|
IF( KS.GE.KBOT ) THEN
|
|
AA = H( KBOT-1, KBOT-1 )
|
|
CC = H( KBOT, KBOT-1 )
|
|
BB = H( KBOT-1, KBOT )
|
|
DD = H( KBOT, KBOT )
|
|
CALL SLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
|
|
$ WI( KBOT-1 ), WR( KBOT ),
|
|
$ WI( KBOT ), CS, SN )
|
|
KS = KBOT - 1
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( KBOT-KS+1.GT.NS ) THEN
|
|
*
|
|
* ==== Sort the shifts (Helps a little)
|
|
* . Bubble sort keeps complex conjugate
|
|
* . pairs together. ====
|
|
*
|
|
SORTED = .false.
|
|
DO 50 K = KBOT, KS + 1, -1
|
|
IF( SORTED )
|
|
$ GO TO 60
|
|
SORTED = .true.
|
|
DO 40 I = KS, K - 1
|
|
IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
|
|
$ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
|
|
SORTED = .false.
|
|
*
|
|
SWAP = WR( I )
|
|
WR( I ) = WR( I+1 )
|
|
WR( I+1 ) = SWAP
|
|
*
|
|
SWAP = WI( I )
|
|
WI( I ) = WI( I+1 )
|
|
WI( I+1 ) = SWAP
|
|
END IF
|
|
40 CONTINUE
|
|
50 CONTINUE
|
|
60 CONTINUE
|
|
END IF
|
|
*
|
|
* ==== Shuffle shifts into pairs of real shifts
|
|
* . and pairs of complex conjugate shifts
|
|
* . assuming complex conjugate shifts are
|
|
* . already adjacent to one another. (Yes,
|
|
* . they are.) ====
|
|
*
|
|
DO 70 I = KBOT, KS + 2, -2
|
|
IF( WI( I ).NE.-WI( I-1 ) ) THEN
|
|
*
|
|
SWAP = WR( I )
|
|
WR( I ) = WR( I-1 )
|
|
WR( I-1 ) = WR( I-2 )
|
|
WR( I-2 ) = SWAP
|
|
*
|
|
SWAP = WI( I )
|
|
WI( I ) = WI( I-1 )
|
|
WI( I-1 ) = WI( I-2 )
|
|
WI( I-2 ) = SWAP
|
|
END IF
|
|
70 CONTINUE
|
|
END IF
|
|
*
|
|
* ==== If there are only two shifts and both are
|
|
* . real, then use only one. ====
|
|
*
|
|
IF( KBOT-KS+1.EQ.2 ) THEN
|
|
IF( WI( KBOT ).EQ.ZERO ) THEN
|
|
IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
|
|
$ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
|
|
WR( KBOT-1 ) = WR( KBOT )
|
|
ELSE
|
|
WR( KBOT ) = WR( KBOT-1 )
|
|
END IF
|
|
END IF
|
|
END IF
|
|
*
|
|
* ==== Use up to NS of the the smallest magnatiude
|
|
* . shifts. If there aren't NS shifts available,
|
|
* . then use them all, possibly dropping one to
|
|
* . make the number of shifts even. ====
|
|
*
|
|
NS = MIN( NS, KBOT-KS+1 )
|
|
NS = NS - MOD( NS, 2 )
|
|
KS = KBOT - NS + 1
|
|
*
|
|
* ==== Small-bulge multi-shift QR sweep:
|
|
* . split workspace under the subdiagonal into
|
|
* . - a KDU-by-KDU work array U in the lower
|
|
* . left-hand-corner,
|
|
* . - a KDU-by-at-least-KDU-but-more-is-better
|
|
* . (KDU-by-NHo) horizontal work array WH along
|
|
* . the bottom edge,
|
|
* . - and an at-least-KDU-but-more-is-better-by-KDU
|
|
* . (NVE-by-KDU) vertical work WV arrow along
|
|
* . the left-hand-edge. ====
|
|
*
|
|
KDU = 3*NS - 3
|
|
KU = N - KDU + 1
|
|
KWH = KDU + 1
|
|
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
|
|
KWV = KDU + 4
|
|
NVE = N - KDU - KWV + 1
|
|
*
|
|
* ==== Small-bulge multi-shift QR sweep ====
|
|
*
|
|
CALL SLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
|
|
$ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
|
|
$ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
|
|
$ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
|
|
END IF
|
|
*
|
|
* ==== Note progress (or the lack of it). ====
|
|
*
|
|
IF( LD.GT.0 ) THEN
|
|
NDFL = 1
|
|
ELSE
|
|
NDFL = NDFL + 1
|
|
END IF
|
|
*
|
|
* ==== End of main loop ====
|
|
80 CONTINUE
|
|
*
|
|
* ==== Iteration limit exceeded. Set INFO to show where
|
|
* . the problem occurred and exit. ====
|
|
*
|
|
INFO = KBOT
|
|
90 CONTINUE
|
|
END IF
|
|
*
|
|
* ==== Return the optimal value of LWORK. ====
|
|
*
|
|
WORK( 1 ) = REAL( LWKOPT )
|
|
*
|
|
* ==== End of SLAQR4 ====
|
|
*
|
|
END
|