/* -- translated by f2c (version 20100827).
You must link the resulting object file with libf2c:
on Microsoft Windows system, link with libf2c.lib;
on Linux or Unix systems, link with .../path/to/libf2c.a -lm
or, if you install libf2c.a in a standard place, with -lf2c -lm
-- in that order, at the end of the command line, as in
cc *.o -lf2c -lm
Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
http://www.netlib.org/f2c/libf2c.zip
*/
#include "f2c.h"
/* Table of constant values */
static integer c__1 = 1;
/* -----------------------------------------------------------------------
\BeginDoc
\Name: dlaqrb
\Description:
Compute the eigenvalues and the Schur decomposition of an upper
Hessenberg submatrix in rows and columns ILO to IHI. Only the
last component of the Schur vectors are computed.
This is mostly a modification of the LAPACK routine dlahqr.
\Usage:
call dlaqrb
( WANTT, N, ILO, IHI, H, LDH, WR, WI, Z, INFO )
\Arguments
WANTT Logical variable. (INPUT)
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.
N Integer. (INPUT)
The order of the matrix H. N >= 0.
ILO Integer. (INPUT)
IHI Integer. (INPUT)
It is assumed that H is already upper quasi-triangular in
rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
ILO = 1). SLAQRB works primarily with the Hessenberg
submatrix in rows and columns ILO to IHI, but applies
transformations to all of H if WANTT is .TRUE..
1 <= ILO <= max(1,IHI); IHI <= N.
H Double precision array, dimension (LDH,N). (INPUT/OUTPUT)
On entry, the upper Hessenberg matrix H.
On exit, if WANTT is .TRUE., H is upper quasi-triangular in
rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in
standard form. If WANTT is .FALSE., the contents of H are
unspecified on exit.
LDH Integer. (INPUT)
The leading dimension of the array H. LDH >= max(1,N).
WR Double precision array, dimension (N). (OUTPUT)
WI Double precision array, dimension (N). (OUTPUT)
The real and imaginary parts, respectively, of the computed
eigenvalues ILO to IHI are stored in the corresponding
elements of WR and WI. 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) > 0 and WI(i+1) < 0. If WANTT is .TRUE., 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 Double precision array, dimension (N). (OUTPUT)
On exit Z contains the last components of the Schur vectors.
INFO Integer. (OUPUT)
= 0: successful exit
> 0: SLAQRB failed to compute all the eigenvalues ILO to IHI
in a total of 30*(IHI-ILO+1) iterations; if INFO = i,
elements i+1:ihi of WR and WI contain those eigenvalues
which have been successfully computed.
\Remarks
1. None.
-----------------------------------------------------------------------
\BeginLib
\Local variables:
xxxxxx real
\Routines called:
dlabad LAPACK routine that computes machine constants.
dlamch LAPACK routine that determines machine constants.
dlanhs LAPACK routine that computes various norms of a matrix.
dlanv2 LAPACK routine that computes the Schur factorization of
2 by 2 nonsymmetric matrix in standard form.
dlarfg LAPACK Householder reflection construction routine.
dcopy Level 1 BLAS that copies one vector to another.
drot Level 1 BLAS that applies a rotation to a 2 by 2 matrix.
\Author
Danny Sorensen Phuong Vu
Richard Lehoucq CRPC / Rice University
Dept. of Computational & Houston, Texas
Applied Mathematics
Rice University
Houston, Texas
\Revision history:
xx/xx/92: Version ' 2.4'
Modified from the LAPACK routine dlahqr so that only the
last component of the Schur vectors are computed.
\SCCS Information: @(#)
FILE: laqrb.F SID: 2.2 DATE OF SID: 8/27/96 RELEASE: 2
\Remarks
1. None
\EndLib
-----------------------------------------------------------------------
Subroutine */ int igraphdlaqrb_(logical *wantt, integer *n, integer *ilo,
integer *ihi, doublereal *h__, integer *ldh, doublereal *wr,
doublereal *wi, doublereal *z__, integer *info)
{
/* System generated locals */
integer h_dim1, h_offset, i__1, i__2, i__3, i__4;
doublereal d__1, d__2;
/* Local variables */
integer i__, j, k, l, m;
doublereal s, v[3];
integer i1, i2;
doublereal t1, t2, t3, v1, v2, v3, h00, h10, h11, h12, h21, h22, h33, h44;
integer nh;
doublereal cs;
integer nr;
doublereal sn, h33s, h44s;
integer itn, its;
doublereal ulp, sum, tst1, h43h34, unfl, ovfl;
extern /* Subroutine */ int igraphdrot_(integer *, doublereal *, integer *,
doublereal *, integer *, doublereal *, doublereal *);
doublereal work[1];
extern /* Subroutine */ int igraphdcopy_(integer *, doublereal *, integer *,
doublereal *, integer *), igraphdlanv2_(doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *), igraphdlabad_(
doublereal *, doublereal *);
extern doublereal igraphdlamch_(char *);
extern /* Subroutine */ int igraphdlarfg_(integer *, doublereal *, doublereal *,
integer *, doublereal *);
extern doublereal igraphdlanhs_(char *, integer *, doublereal *, integer *,
doublereal *);
doublereal smlnum;
/* %------------------%
| Scalar Arguments |
%------------------%
%-----------------%
| Array Arguments |
%-----------------%
%------------%
| Parameters |
%------------%
%------------------------%
| Local Scalars & Arrays |
%------------------------%
%--------------------%
| External Functions |
%--------------------%
%----------------------%
| External Subroutines |
%----------------------%
%-----------------------%
| Executable Statements |
%-----------------------%
Parameter adjustments */
h_dim1 = *ldh;
h_offset = 1 + h_dim1;
h__ -= h_offset;
--wr;
--wi;
--z__;
/* Function Body */
*info = 0;
/* %--------------------------%
| Quick return if possible |
%--------------------------% */
if (*n == 0) {
return 0;
}
if (*ilo == *ihi) {
wr[*ilo] = h__[*ilo + *ilo * h_dim1];
wi[*ilo] = 0.;
return 0;
}
/* %---------------------------------------------%
| Initialize the vector of last components of |
| the Schur vectors for accumulation. |
%---------------------------------------------% */
i__1 = *n - 1;
for (j = 1; j <= i__1; ++j) {
z__[j] = 0.;
/* L5: */
}
z__[*n] = 1.;
nh = *ihi - *ilo + 1;
/* %-------------------------------------------------------------%
| Set machine-dependent constants for the stopping criterion. |
| If norm(H) <= sqrt(OVFL), overflow should not occur. |
%-------------------------------------------------------------% */
unfl = igraphdlamch_("safe minimum");
ovfl = 1. / unfl;
igraphdlabad_(&unfl, &ovfl);
ulp = igraphdlamch_("precision");
smlnum = unfl * (nh / ulp);
/* %---------------------------------------------------------------%
| I1 and I2 are the indices of the first row and last column |
| of H to which transformations must be applied. If eigenvalues |
| only are computed, I1 and I2 are set inside the main loop. |
| Zero out H(J+2,J) = ZERO for J=1:N if WANTT = .TRUE. |
| else H(J+2,J) for J=ILO:IHI-ILO-1 if WANTT = .FALSE. |
%---------------------------------------------------------------% */
if (*wantt) {
i1 = 1;
i2 = *n;
i__1 = i2 - 2;
for (i__ = 1; i__ <= i__1; ++i__) {
h__[i1 + i__ + 1 + i__ * h_dim1] = 0.;
/* L8: */
}
} else {
i__1 = *ihi - *ilo - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
h__[*ilo + i__ + 1 + (*ilo + i__ - 1) * h_dim1] = 0.;
/* L9: */
}
}
/* %---------------------------------------------------%
| ITN is the total number of QR iterations allowed. |
%---------------------------------------------------% */
itn = nh * 30;
/* ------------------------------------------------------------------
The main loop begins here. I is the loop index and decreases from
IHI to ILO in steps of 1 or 2. Each iteration of the loop works
with the active submatrix in rows and columns L to I.
Eigenvalues I+1 to IHI have already converged. Either L = ILO or
H(L,L-1) is negligible so that the matrix splits.
------------------------------------------------------------------ */
i__ = *ihi;
L10:
l = *ilo;
if (i__ < *ilo) {
goto L150;
}
/* %--------------------------------------------------------------%
| Perform QR iterations on rows and columns ILO to I until a |
| submatrix of order 1 or 2 splits off at the bottom because a |
| subdiagonal element has become negligible. |
%--------------------------------------------------------------% */
i__1 = itn;
for (its = 0; its <= i__1; ++its) {
/* %----------------------------------------------%
| Look for a single small subdiagonal element. |
%----------------------------------------------% */
i__2 = l + 1;
for (k = i__; k >= i__2; --k) {
tst1 = (d__1 = h__[k - 1 + (k - 1) * h_dim1], abs(d__1)) + (d__2 =
h__[k + k * h_dim1], abs(d__2));
if (tst1 == 0.) {
i__3 = i__ - l + 1;
tst1 = igraphdlanhs_("1", &i__3, &h__[l + l * h_dim1], ldh, work);
}
/* Computing MAX */
d__2 = ulp * tst1;
if ((d__1 = h__[k + (k - 1) * h_dim1], abs(d__1)) <= max(d__2,
smlnum)) {
goto L30;
}
/* L20: */
}
L30:
l = k;
if (l > *ilo) {
/* %------------------------%
| H(L,L-1) is negligible |
%------------------------% */
h__[l + (l - 1) * h_dim1] = 0.;
}
/* %-------------------------------------------------------------%
| Exit from loop if a submatrix of order 1 or 2 has split off |
%-------------------------------------------------------------% */
if (l >= i__ - 1) {
goto L140;
}
/* %---------------------------------------------------------%
| Now the active submatrix is in rows and columns L to I. |
| If eigenvalues only are being computed, only the active |
| submatrix need be transformed. |
%---------------------------------------------------------% */
if (! (*wantt)) {
i1 = l;
i2 = i__;
}
if (its == 10 || its == 20) {
/* %-------------------%
| Exceptional shift |
%-------------------% */
s = (d__1 = h__[i__ + (i__ - 1) * h_dim1], abs(d__1)) + (d__2 =
h__[i__ - 1 + (i__ - 2) * h_dim1], abs(d__2));
h44 = s * .75;
h33 = h44;
h43h34 = s * -.4375 * s;
} else {
/* %-----------------------------------------%
| Prepare to use Wilkinson's double shift |
%-----------------------------------------% */
h44 = h__[i__ + i__ * h_dim1];
h33 = h__[i__ - 1 + (i__ - 1) * h_dim1];
h43h34 = h__[i__ + (i__ - 1) * h_dim1] * h__[i__ - 1 + i__ *
h_dim1];
}
/* %-----------------------------------------------------%
| Look for two consecutive small subdiagonal elements |
%-----------------------------------------------------% */
i__2 = l;
for (m = i__ - 2; m >= i__2; --m) {
/* %---------------------------------------------------------%
| Determine the effect of starting the double-shift QR |
| iteration at row M, and see if this would make H(M,M-1) |
| negligible. |
%---------------------------------------------------------% */
h11 = h__[m + m * h_dim1];
h22 = h__[m + 1 + (m + 1) * h_dim1];
h21 = h__[m + 1 + m * h_dim1];
h12 = h__[m + (m + 1) * h_dim1];
h44s = h44 - h11;
h33s = h33 - h11;
v1 = (h33s * h44s - h43h34) / h21 + h12;
v2 = h22 - h11 - h33s - h44s;
v3 = h__[m + 2 + (m + 1) * h_dim1];
s = abs(v1) + abs(v2) + abs(v3);
v1 /= s;
v2 /= s;
v3 /= s;
v[0] = v1;
v[1] = v2;
v[2] = v3;
if (m == l) {
goto L50;
}
h00 = h__[m - 1 + (m - 1) * h_dim1];
h10 = h__[m + (m - 1) * h_dim1];
tst1 = abs(v1) * (abs(h00) + abs(h11) + abs(h22));
if (abs(h10) * (abs(v2) + abs(v3)) <= ulp * tst1) {
goto L50;
}
/* L40: */
}
L50:
/* %----------------------%
| Double-shift QR step |
%----------------------% */
i__2 = i__ - 1;
for (k = m; k <= i__2; ++k) {
/* ------------------------------------------------------------
The first iteration of this loop determines a reflection G
from the vector V and applies it from left and right to H,
thus creating a nonzero bulge below the subdiagonal.
Each subsequent iteration determines a reflection G to
restore the Hessenberg form in the (K-1)th column, and thus
chases the bulge one step toward the bottom of the active
submatrix. NR is the order of G.
------------------------------------------------------------
Computing MIN */
i__3 = 3, i__4 = i__ - k + 1;
nr = min(i__3,i__4);
if (k > m) {
igraphdcopy_(&nr, &h__[k + (k - 1) * h_dim1], &c__1, v, &c__1);
}
igraphdlarfg_(&nr, v, &v[1], &c__1, &t1);
if (k > m) {
h__[k + (k - 1) * h_dim1] = v[0];
h__[k + 1 + (k - 1) * h_dim1] = 0.;
if (k < i__ - 1) {
h__[k + 2 + (k - 1) * h_dim1] = 0.;
}
} else if (m > l) {
h__[k + (k - 1) * h_dim1] = -h__[k + (k - 1) * h_dim1];
}
v2 = v[1];
t2 = t1 * v2;
if (nr == 3) {
v3 = v[2];
t3 = t1 * v3;
/* %------------------------------------------------%
| Apply G from the left to transform the rows of |
| the matrix in columns K to I2. |
%------------------------------------------------% */
i__3 = i2;
for (j = k; j <= i__3; ++j) {
sum = h__[k + j * h_dim1] + v2 * h__[k + 1 + j * h_dim1]
+ v3 * h__[k + 2 + j * h_dim1];
h__[k + j * h_dim1] -= sum * t1;
h__[k + 1 + j * h_dim1] -= sum * t2;
h__[k + 2 + j * h_dim1] -= sum * t3;
/* L60: */
}
/* %----------------------------------------------------%
| Apply G from the right to transform the columns of |
| the matrix in rows I1 to min(K+3,I). |
%----------------------------------------------------%
Computing MIN */
i__4 = k + 3;
i__3 = min(i__4,i__);
for (j = i1; j <= i__3; ++j) {
sum = h__[j + k * h_dim1] + v2 * h__[j + (k + 1) * h_dim1]
+ v3 * h__[j + (k + 2) * h_dim1];
h__[j + k * h_dim1] -= sum * t1;
h__[j + (k + 1) * h_dim1] -= sum * t2;
h__[j + (k + 2) * h_dim1] -= sum * t3;
/* L70: */
}
/* %----------------------------------%
| Accumulate transformations for Z |
%----------------------------------% */
sum = z__[k] + v2 * z__[k + 1] + v3 * z__[k + 2];
z__[k] -= sum * t1;
z__[k + 1] -= sum * t2;
z__[k + 2] -= sum * t3;
} else if (nr == 2) {
/* %------------------------------------------------%
| Apply G from the left to transform the rows of |
| the matrix in columns K to I2. |
%------------------------------------------------% */
i__3 = i2;
for (j = k; j <= i__3; ++j) {
sum = h__[k + j * h_dim1] + v2 * h__[k + 1 + j * h_dim1];
h__[k + j * h_dim1] -= sum * t1;
h__[k + 1 + j * h_dim1] -= sum * t2;
/* L90: */
}
/* %----------------------------------------------------%
| Apply G from the right to transform the columns of |
| the matrix in rows I1 to min(K+3,I). |
%----------------------------------------------------% */
i__3 = i__;
for (j = i1; j <= i__3; ++j) {
sum = h__[j + k * h_dim1] + v2 * h__[j + (k + 1) * h_dim1]
;
h__[j + k * h_dim1] -= sum * t1;
h__[j + (k + 1) * h_dim1] -= sum * t2;
/* L100: */
}
/* %----------------------------------%
| Accumulate transformations for Z |
%----------------------------------% */
sum = z__[k] + v2 * z__[k + 1];
z__[k] -= sum * t1;
z__[k + 1] -= sum * t2;
}
/* L120: */
}
/* L130: */
}
/* %-------------------------------------------------------%
| Failure to converge in remaining number of iterations |
%-------------------------------------------------------% */
*info = i__;
return 0;
L140:
if (l == i__) {
/* %------------------------------------------------------%
| H(I,I-1) is negligible: one eigenvalue has converged |
%------------------------------------------------------% */
wr[i__] = h__[i__ + i__ * h_dim1];
wi[i__] = 0.;
} else if (l == i__ - 1) {
/* %--------------------------------------------------------%
| H(I-1,I-2) is negligible; |
| a pair of eigenvalues have converged. |
| |
| Transform the 2-by-2 submatrix to standard Schur form, |
| and compute and store the eigenvalues. |
%--------------------------------------------------------% */
igraphdlanv2_(&h__[i__ - 1 + (i__ - 1) * h_dim1], &h__[i__ - 1 + i__ *
h_dim1], &h__[i__ + (i__ - 1) * h_dim1], &h__[i__ + i__ *
h_dim1], &wr[i__ - 1], &wi[i__ - 1], &wr[i__], &wi[i__], &cs,
&sn);
if (*wantt) {
/* %-----------------------------------------------------%
| Apply the transformation to the rest of H and to Z, |
| as required. |
%-----------------------------------------------------% */
if (i2 > i__) {
i__1 = i2 - i__;
igraphdrot_(&i__1, &h__[i__ - 1 + (i__ + 1) * h_dim1], ldh, &h__[
i__ + (i__ + 1) * h_dim1], ldh, &cs, &sn);
}
i__1 = i__ - i1 - 1;
igraphdrot_(&i__1, &h__[i1 + (i__ - 1) * h_dim1], &c__1, &h__[i1 + i__ *
h_dim1], &c__1, &cs, &sn);
sum = cs * z__[i__ - 1] + sn * z__[i__];
z__[i__] = cs * z__[i__] - sn * z__[i__ - 1];
z__[i__ - 1] = sum;
}
}
/* %---------------------------------------------------------%
| Decrement number of remaining iterations, and return to |
| start of the main loop with new value of I. |
%---------------------------------------------------------% */
itn -= its;
i__ = l - 1;
goto L10;
L150:
return 0;
/* %---------------%
| End of dlaqrb |
%---------------% */
} /* igraphdlaqrb_ */