#include "blaswrap.h"
#include "f2c.h"
/* Subroutine */ int shseqr_(char *job, char *compz, integer *n, integer *ilo,
integer *ihi, real *h__, integer *ldh, real *wr, real *wi, real *z__, integer *ldz, real *work, integer *lwork, integer *info )
{
/* -- LAPACK driver routine (version 3.1) --
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
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) CHARACTER*1
= 'E': compute eigenvalues only;
= 'S': compute eigenvalues and the Schur form T.
COMPZ (input) 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) INTEGER
The order of the matrix H. N .GE. 0.
ILO (input) INTEGER
IHI (input) 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 SGEHRD
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) 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) INTEGER
The leading dimension of the array H. LDH .GE. max(1,N).
WR (output) REAL array, dimension (N)
WI (output) 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) 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) 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 (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns an estimate of
the optimal value for LWORK.
LWORK (input) 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 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) 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.
================================================================
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=1: The SLAHQR vs SLAQR0 crossover point.
Default: 75. (Must be at least 11.)
ISPEC=2: Recommended deflation window size.
This depends on ILO, IHI and NS. NS is the
number of simultaneous shifts returned
by ILAENV(ISPEC=4). (See ISPEC=4 below.)
The default for (IHI-ILO+1).LE.500 is NS.
The default for (IHI-ILO+1).GT.500 is 3*NS/2.
ISPEC=3: Nibble crossover point. (See ILAENV for
details.) Default: 14% of deflation window
size.
ISPEC=4: Number of simultaneous shifts, NS, in
a multi-shift 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 NS is ignored. See ISPEC=1 above
and comments in IPARM for details.
The asterisks (**) indicate an ad-hoc
function of N increasing from 10 to 64.
ISPEC=5: Select structured matrix multiply.
(See ILAENV for details.) Default: 3.
================================================================
Based on contributions by
Karen Braman and Ralph Byers, Department of Mathematics,
University of Kansas, USA
================================================================
References:
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
Performance, SIAM Journal of Matrix Analysis, volume 23, pages
929--947, 2002.
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
Algorithm Part II: Aggressive Early Deflation, SIAM Journal
of Matrix Analysis, volume 23, pages 948--973, 2002.
================================================================
==== Matrices of order NTINY or smaller must be processed by
. SLAHQR because of insufficient subdiagonal scratch space.
. (This is a hard limit.) ====
==== NL allocates some local workspace to help small matrices
. through a rare SLAHQR failure. NL .GT. NTINY = 11 is
. required and NL .LE. NMIN = ILAENV(ISPEC=1,...) is recom-
. mended. (The default value of NMIN is 75.) Using NL = 49
. allows up to six simultaneous shifts and a 16-by-16
. deflation window. ====
==== Decode and check the input parameters. ====
Parameter adjustments */
/* Table of constant values */
static real c_b11 = 0.f;
static real c_b12 = 1.f;
static integer c__1 = 1;
static integer c__2 = 2;
static integer c__49 = 49;
/* System generated locals */
address a__1[2];
integer h_dim1, h_offset, z_dim1, z_offset, i__1, i__2[2], i__3;
real r__1;
char ch__1[2];
/* Builtin functions
Subroutine */ int s_cat(char *, char **, integer *, integer *, ftnlen);
/* Local variables */
static integer i__;
static real hl[2401] /* was [49][49] */;
static integer kbot, nmin;
extern logical lsame_(char *, char *);
static logical initz;
static real workl[49];
static logical wantt, wantz;
extern /* Subroutine */ int slaqr0_(logical *, logical *, integer *,
integer *, integer *, real *, integer *, real *, real *, integer *
, integer *, real *, integer *, real *, integer *, integer *),
xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern /* Subroutine */ int slahqr_(logical *, logical *, integer *,
integer *, integer *, real *, integer *, real *, real *, integer *
, integer *, real *, integer *, integer *), slacpy_(char *,
integer *, integer *, real *, integer *, real *, integer *), slaset_(char *, integer *, integer *, real *, real *,
real *, integer *);
static logical lquery;
h_dim1 = *ldh;
h_offset = 1 + h_dim1;
h__ -= h_offset;
--wr;
--wi;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
/* Function Body */
wantt = lsame_(job, "S");
initz = lsame_(compz, "I");
wantz = initz || lsame_(compz, "V");
work[1] = (real) max(1,*n);
lquery = *lwork == -1;
*info = 0;
if (! lsame_(job, "E") && ! wantt) {
*info = -1;
} else if (! lsame_(compz, "N") && ! wantz) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*ilo < 1 || *ilo > max(1,*n)) {
*info = -4;
} else if (*ihi < min(*ilo,*n) || *ihi > *n) {
*info = -5;
} else if (*ldh < max(1,*n)) {
*info = -7;
} else if (*ldz < 1 || wantz && *ldz < max(1,*n)) {
*info = -11;
} else if (*lwork < max(1,*n) && ! lquery) {
*info = -13;
}
if (*info != 0) {
/* ==== Quick return in case of invalid argument. ==== */
i__1 = -(*info);
xerbla_("SHSEQR", &i__1);
return 0;
} else if (*n == 0) {
/* ==== Quick return in case N = 0; nothing to do. ==== */
return 0;
} else if (lquery) {
/* ==== Quick return in case of a workspace query ==== */
slaqr0_(&wantt, &wantz, n, ilo, ihi, &h__[h_offset], ldh, &wr[1], &wi[
1], ilo, ihi, &z__[z_offset], ldz, &work[1], lwork, info);
/* ==== Ensure reported workspace size is backward-compatible with
. previous LAPACK versions. ====
Computing MAX */
r__1 = (real) max(1,*n);
work[1] = dmax(r__1,work[1]);
return 0;
} else {
/* ==== copy eigenvalues isolated by SGEBAL ==== */
i__1 = *ilo - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
wr[i__] = h__[i__ + i__ * h_dim1];
wi[i__] = 0.f;
/* L10: */
}
i__1 = *n;
for (i__ = *ihi + 1; i__ <= i__1; ++i__) {
wr[i__] = h__[i__ + i__ * h_dim1];
wi[i__] = 0.f;
/* L20: */
}
/* ==== Initialize Z, if requested ==== */
if (initz) {
slaset_("A", n, n, &c_b11, &c_b12, &z__[z_offset], ldz)
;
}
/* ==== Quick return if possible ==== */
if (*ilo == *ihi) {
wr[*ilo] = h__[*ilo + *ilo * h_dim1];
wi[*ilo] = 0.f;
return 0;
}
/* ==== SLAHQR/SLAQR0 crossover point ====
Writing concatenation */
i__2[0] = 1, a__1[0] = job;
i__2[1] = 1, a__1[1] = compz;
s_cat(ch__1, a__1, i__2, &c__2, (ftnlen)2);
nmin = ilaenv_(&c__1, "SHSEQR", ch__1, n, ilo, ihi, lwork, (ftnlen)6,
(ftnlen)2);
nmin = max(11,nmin);
/* ==== SLAQR0 for big matrices; SLAHQR for small ones ==== */
if (*n > nmin) {
slaqr0_(&wantt, &wantz, n, ilo, ihi, &h__[h_offset], ldh, &wr[1],
&wi[1], ilo, ihi, &z__[z_offset], ldz, &work[1], lwork,
info);
} else {
/* ==== Small matrix ==== */
slahqr_(&wantt, &wantz, n, ilo, ihi, &h__[h_offset], ldh, &wr[1],
&wi[1], ilo, ihi, &z__[z_offset], ldz, info);
if (*info > 0) {
/* ==== A rare SLAHQR failure! SLAQR0 sometimes succeeds
. when SLAHQR fails. ==== */
kbot = *info;
if (*n >= 49) {
/* ==== Larger matrices have enough subdiagonal scratch
. space to call SLAQR0 directly. ==== */
slaqr0_(&wantt, &wantz, n, ilo, &kbot, &h__[h_offset],
ldh, &wr[1], &wi[1], ilo, ihi, &z__[z_offset],
ldz, &work[1], lwork, info);
} else {
/* ==== Tiny matrices don't have enough subdiagonal
. scratch space to benefit from SLAQR0. Hence,
. tiny matrices must be copied into a larger
. array before calling SLAQR0. ==== */
slacpy_("A", n, n, &h__[h_offset], ldh, hl, &c__49);
hl[*n + 1 + *n * 49 - 50] = 0.f;
i__1 = 49 - *n;
slaset_("A", &c__49, &i__1, &c_b11, &c_b11, &hl[(*n + 1) *
49 - 49], &c__49);
slaqr0_(&wantt, &wantz, &c__49, ilo, &kbot, hl, &c__49, &
wr[1], &wi[1], ilo, ihi, &z__[z_offset], ldz,
workl, &c__49, info);
if (wantt || *info != 0) {
slacpy_("A", n, n, hl, &c__49, &h__[h_offset], ldh);
}
}
}
}
/* ==== Clear out the trash, if necessary. ==== */
if ((wantt || *info != 0) && *n > 2) {
i__1 = *n - 2;
i__3 = *n - 2;
slaset_("L", &i__1, &i__3, &c_b11, &c_b11, &h__[h_dim1 + 3], ldh);
}
/* ==== Ensure reported workspace size is backward-compatible with
. previous LAPACK versions. ====
Computing MAX */
r__1 = (real) max(1,*n);
work[1] = dmax(r__1,work[1]);
}
/* ==== End of SHSEQR ==== */
return 0;
} /* shseqr_ */