DLAQR4 (3)  Linux Manuals
NAME
dlaqr4.f 
SYNOPSIS
Functions/Subroutines
subroutine dlaqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
Function/Subroutine Documentation
subroutine dlaqr4 (logicalWANTT, logicalWANTZ, integerN, integerILO, integerIHI, double precision, dimension( ldh, * )H, integerLDH, double precision, dimension( * )WR, double precision, dimension( * )WI, integerILOZ, integerIHIZ, double precision, dimension( ldz, * )Z, integerLDZ, double precision, dimension( * )WORK, integerLWORK, integerINFO)
DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
Purpose:

DLAQR4 implements one level of recursion for DLAQR0. It is a complete implementation of the small bulge multishift QR algorithm. It may be called by DLAQR0 and, for large enough deflation window size, it may be called by DLAQR3. This subroutine is identical to DLAQR0 except that it calls DLAQR2 instead of DLAQR3. DLAQR4 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 quasitriangular 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.
Parameters:

WANTT
WANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required.
WANTZWANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required.
NN is INTEGER The order of the matrix H. N .GE. 0.
ILOILO is INTEGER
IHIIHI is INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO1 and IHI+1:N and, if ILO.GT.1, H(ILO,ILO1) is zero. ILO and IHI are normally set by a previous call to DGEBAL, and then passed to DGEHRD when the matrix output by DGEBAL 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.
HH is DOUBLE PRECISION array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and WANTT is .TRUE., then H contains the upper quasitriangular matrix T from the Schur decomposition (the Schur form); 2by2 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 WANTT is .FALSE., then 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.) This subroutine may explicitly set H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO1 or j = IHI+1, IHI+2, ... N.
LDHLDH is INTEGER The leading dimension of the array H. LDH .GE. max(1,N).
WRWR is DOUBLE PRECISION array, dimension (IHI)
WIWI is DOUBLE PRECISION array, dimension (IHI) The real and imaginary parts, respectively, of the computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) and WI(ILO:IHI). If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the ith and (i+1)th, with WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then 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 2by2 diagonal block, WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = WI(i).
ILOZILOZ is INTEGER
IHIZIHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
ZZ is DOUBLE PRECISION array, dimension (LDZ,IHI) If WANTZ is .FALSE., then Z is not referenced. If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value of Z when INFO.GT.0 is given under the description of INFO below.)
LDZLDZ is INTEGER The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
WORKWORK is DOUBLE PRECISION array, dimension LWORK On exit, if LWORK = 1, WORK(1) returns an estimate of the optimal value for LWORK.
LWORKLWORK is INTEGER The dimension of the array WORK. LWORK .GE. max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. If LWORK = 1, then DLAQR4 does a workspace query. In this case, DLAQR4 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.
INFOINFO is INTEGER = 0: successful exit .GT. 0: if INFO = i, DLAQR4 failed to compute all of the eigenvalues. Elements 1:ilo1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO .GT. 0 and WANT is .FALSE., 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 WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is a orthogonal matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI. If INFO .GT. 0 and WANTZ is .TRUE., then on exit (final value of Z(ILO:IHI,ILOZ:IHIZ) = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal matrix in (*) (regard less of the value of WANTT.) If INFO .GT. 0 and WANTZ is .FALSE., then Z is not accessed.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 September 2012
Contributors:
 Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
References:

K. Braman, R. Byers and R. Mathias, The MultiShift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929947, 2002.
K. Braman, R. Byers and R. Mathias, The MultiShift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948973, 2002.
Definition at line 263 of file dlaqr4.f.
Author
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