shseqr (3)  Linux Manuals
NAME
shseqr.f 
SYNOPSIS
Functions/Subroutines
subroutine shseqr (JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK, LWORK, INFO)
SHSEQR
Function/Subroutine Documentation
subroutine shseqr (characterJOB, characterCOMPZ, integerN, integerILO, integerIHI, real, dimension( ldh, * )H, integerLDH, real, dimension( * )WR, real, dimension( * )WI, real, dimension( ldz, * )Z, integerLDZ, real, dimension( * )WORK, integerLWORK, integerINFO)
SHSEQR
Purpose:

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 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:

JOB
JOB is CHARACTER*1 = 'E': compute eigenvalues only; = 'S': compute eigenvalues and the Schur form T.
COMPZCOMPZ 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.
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. 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.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.
HH 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 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 JOB = 'E', 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.) Unlike earlier versions of SHSEQR, this subroutine may explicitly 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 REAL array, dimension (N)
WIWI 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 ith and (i+1)th, with WI(i) .GT. 0 and WI(i+1) .LT. 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 2by2 diagonal block, WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = WI(i).
ZZ 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 NbyN 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.GT.0 is given under the description of INFO below.)
LDZLDZ is INTEGER The leading dimension of the array Z. if COMPZ = 'I' or COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.
WORKWORK is REAL array, dimension (LWORK) On exit, if INFO = 0, 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 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.
INFOINFO is INTEGER = 0: successful exit .LT. 0: if INFO = i, the ith argument had an illegal value .GT. 0: if INFO = i, SHSEQR 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 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 .GT. 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 quasitriangular in rows and columns INFO+1 through IHI. If INFO .GT. 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 .GT. 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 .GT. 0 and COMPZ = 'N', then Z is not accessed.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
Contributors:
 Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
Further Details:

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 (IHIILO+1).LE.500 is NS. The default for (IHIILO+1).GT.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 IHIILO+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 adhoc 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.
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 316 of file shseqr.f.
Author
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