SHSEQR(3) LAPACK routine of NEC Numeric Library Collection SHSEQR(3)
NAME
SHSEQR
SYNOPSIS
SUBROUTINE SHSEQR (JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ,
WORK, LWORK, INFO)
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 quasi-triangular 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.
ARGUMENTS
JOB (input)
JOB is CHARACTER*1
= 'E': compute eigenvalues only;
= 'S': compute eigenvalues and the Schur form T.
COMPZ (input)
COMPZ 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.
N (input)
N is INTEGER
The order of the matrix H. N .GE. 0.
ILO (input)
ILO is INTEGER
IHI (input)
IHI is INTEGER
It is assumed that H is already upper triangular in rows
and columns 1:ILO-1 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.
H (input/output)
H 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 quasi-triangular matrix T from the Schur decomposition
(the Schur form); 2-by-2 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, ... ILO-1
or j = IHI+1, IHI+2, ... N.
LDH (input)
LDH is INTEGER
The leading dimension of the array H. LDH .GE. max(1,N).
WR (output)
WR is REAL array, dimension (N)
WI (output)
WI 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 i-th 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 2-by-2
diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
WI(i+1) = -WI(i).
Z (input/output)
Z 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
N-by-N 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.)
LDZ (input)
LDZ 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.
WORK (output)
WORK is REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns an estimate of
the optimal value for LWORK.
LWORK (input)
LWORK 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.
INFO (output)
INFO is INTEGER
= 0: successful exit
.LT. 0: if INFO = -i, the i-th argument had an illegal
value
.GT. 0: if INFO = i, SHSEQR failed to compute all of
the eigenvalues. Elements 1:ilo-1 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 quasi-triangular
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.
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 (IHI-ILO+1).LE.500 is NS.
The default for (IHI-ILO+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 IHI-ILO+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 ad-hoc
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.
LAPACK routine 31 October 2017 SHSEQR(3)