/* comqr.f -- translated by f2c (version 19961017).
You must link the resulting object file with the libraries:
-lf2c -lm (in that order)
*/
#include "f2c.h"
/* Subroutine */ int comqr_(integer *nm, integer *n, integer *low, integer *
igh, doublereal *hr, doublereal *hi, doublereal *wr, doublereal *wi,
integer *ierr)
{
/* System generated locals */
integer hr_dim1, hr_offset, hi_dim1, hi_offset, i__1, i__2;
doublereal d__1, d__2, d__3, d__4;
/* Local variables */
extern /* Subroutine */ int cdiv_(doublereal *, doublereal *, doublereal *
, doublereal *, doublereal *, doublereal *);
static doublereal norm;
static integer i__, j, l, en, ll;
static doublereal si, ti, xi, yi, sr, tr, xr, yr;
extern doublereal pythag_(doublereal *, doublereal *);
extern /* Subroutine */ int csroot_(doublereal *, doublereal *,
doublereal *, doublereal *);
static integer lp1, itn, its;
static doublereal zzi, zzr;
static integer enm1;
static doublereal tst1, tst2;
/* THIS SUBROUTINE IS A TRANSLATION OF A UNITARY ANALOGUE OF THE */
/* ALGOL PROCEDURE COMLR, NUM. MATH. 12, 369-376(1968) BY MARTIN */
/* AND WILKINSON. */
/* HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 396-403(1971). */
/* THE UNITARY ANALOGUE SUBSTITUTES THE QR ALGORITHM OF FRANCIS */
/* (COMP. JOUR. 4, 332-345(1962)) FOR THE LR ALGORITHM. */
/* THIS SUBROUTINE FINDS THE EIGENVALUES OF A COMPLEX */
/* UPPER HESSENBERG MATRIX BY THE QR METHOD. */
/* ON INPUT */
/* NM MUST BE SET TO THE ROW DIMENSION OF TWO-DIMENSIONAL */
/* ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM */
/* DIMENSION STATEMENT. */
/* N IS THE ORDER OF THE MATRIX. */
/* LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING */
/* SUBROUTINE CBAL. IF CBAL HAS NOT BEEN USED, */
/* SET LOW=1, IGH=N. */
/* HR AND HI CONTAIN THE REAL AND IMAGINARY PARTS, */
/* RESPECTIVELY, OF THE COMPLEX UPPER HESSENBERG MATRIX. */
/* THEIR LOWER TRIANGLES BELOW THE SUBDIAGONAL CONTAIN */
/* INFORMATION ABOUT THE UNITARY TRANSFORMATIONS USED IN */
/* THE REDUCTION BY CORTH, IF PERFORMED. */
/* ON OUTPUT */
/* THE UPPER HESSENBERG PORTIONS OF HR AND HI HAVE BEEN */
/* DESTROYED. THEREFORE, THEY MUST BE SAVED BEFORE */
/* CALLING COMQR IF SUBSEQUENT CALCULATION OF */
/* EIGENVECTORS IS TO BE PERFORMED. */
/* WR AND WI CONTAIN THE REAL AND IMAGINARY PARTS, */
/* RESPECTIVELY, OF THE EIGENVALUES. IF AN ERROR */
/* EXIT IS MADE, THE EIGENVALUES SHOULD BE CORRECT */
/* FOR INDICES IERR+1,...,N. */
/* IERR IS SET TO */
/* ZERO FOR NORMAL RETURN, */
/* J IF THE LIMIT OF 30*N ITERATIONS IS EXHAUSTED */
/* WHILE THE J-TH EIGENVALUE IS BEING SOUGHT. */
/* CALLS CDIV FOR COMPLEX DIVISION. */
/* CALLS CSROOT FOR COMPLEX SQUARE ROOT. */
/* CALLS PYTHAG FOR DSQRT(A*A + B*B) . */
/* QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW, */
/* MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
*/
/* THIS VERSION DATED AUGUST 1983. */
/* ------------------------------------------------------------------
*/
/* Parameter adjustments */
--wi;
--wr;
hi_dim1 = *nm;
hi_offset = hi_dim1 + 1;
hi -= hi_offset;
hr_dim1 = *nm;
hr_offset = hr_dim1 + 1;
hr -= hr_offset;
/* Function Body */
*ierr = 0;
if (*low == *igh) {
goto L180;
}
/* .......... CREATE REAL SUBDIAGONAL ELEMENTS .......... */
l = *low + 1;
i__1 = *igh;
for (i__ = l; i__ <= i__1; ++i__) {
/* Computing MIN */
i__2 = i__ + 1;
ll = min(i__2,*igh);
if (hi[i__ + (i__ - 1) * hi_dim1] == 0.) {
goto L170;
}
norm = pythag_(&hr[i__ + (i__ - 1) * hr_dim1], &hi[i__ + (i__ - 1) *
hi_dim1]);
yr = hr[i__ + (i__ - 1) * hr_dim1] / norm;
yi = hi[i__ + (i__ - 1) * hi_dim1] / norm;
hr[i__ + (i__ - 1) * hr_dim1] = norm;
hi[i__ + (i__ - 1) * hi_dim1] = 0.;
i__2 = *igh;
for (j = i__; j <= i__2; ++j) {
si = yr * hi[i__ + j * hi_dim1] - yi * hr[i__ + j * hr_dim1];
hr[i__ + j * hr_dim1] = yr * hr[i__ + j * hr_dim1] + yi * hi[i__
+ j * hi_dim1];
hi[i__ + j * hi_dim1] = si;
/* L155: */
}
i__2 = ll;
for (j = *low; j <= i__2; ++j) {
si = yr * hi[j + i__ * hi_dim1] + yi * hr[j + i__ * hr_dim1];
hr[j + i__ * hr_dim1] = yr * hr[j + i__ * hr_dim1] - yi * hi[j +
i__ * hi_dim1];
hi[j + i__ * hi_dim1] = si;
/* L160: */
}
L170:
;
}
/* .......... STORE ROOTS ISOLATED BY CBAL .......... */
L180:
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (i__ >= *low && i__ <= *igh) {
goto L200;
}
wr[i__] = hr[i__ + i__ * hr_dim1];
wi[i__] = hi[i__ + i__ * hi_dim1];
L200:
;
}
en = *igh;
tr = 0.;
ti = 0.;
itn = *n * 30;
/* .......... SEARCH FOR NEXT EIGENVALUE .......... */
L220:
if (en < *low) {
goto L1001;
}
its = 0;
enm1 = en - 1;
/* .......... LOOK FOR SINGLE SMALL SUB-DIAGONAL ELEMENT */
/* FOR L=EN STEP -1 UNTIL LOW D0 -- .......... */
L240:
i__1 = en;
for (ll = *low; ll <= i__1; ++ll) {
l = en + *low - ll;
if (l == *low) {
goto L300;
}
tst1 = (d__1 = hr[l - 1 + (l - 1) * hr_dim1], abs(d__1)) + (d__2 = hi[
l - 1 + (l - 1) * hi_dim1], abs(d__2)) + (d__3 = hr[l + l *
hr_dim1], abs(d__3)) + (d__4 = hi[l + l * hi_dim1], abs(d__4))
;
tst2 = tst1 + (d__1 = hr[l + (l - 1) * hr_dim1], abs(d__1));
if (tst2 == tst1) {
goto L300;
}
/* L260: */
}
/* .......... FORM SHIFT .......... */
L300:
if (l == en) {
goto L660;
}
if (itn == 0) {
goto L1000;
}
if (its == 10 || its == 20) {
goto L320;
}
sr = hr[en + en * hr_dim1];
si = hi[en + en * hi_dim1];
xr = hr[enm1 + en * hr_dim1] * hr[en + enm1 * hr_dim1];
xi = hi[enm1 + en * hi_dim1] * hr[en + enm1 * hr_dim1];
if (xr == 0. && xi == 0.) {
goto L340;
}
yr = (hr[enm1 + enm1 * hr_dim1] - sr) / 2.;
yi = (hi[enm1 + enm1 * hi_dim1] - si) / 2.;
/* Computing 2nd power */
d__2 = yr;
/* Computing 2nd power */
d__3 = yi;
d__1 = d__2 * d__2 - d__3 * d__3 + xr;
d__4 = yr * 2. * yi + xi;
csroot_(&d__1, &d__4, &zzr, &zzi);
if (yr * zzr + yi * zzi >= 0.) {
goto L310;
}
zzr = -zzr;
zzi = -zzi;
L310:
d__1 = yr + zzr;
d__2 = yi + zzi;
cdiv_(&xr, &xi, &d__1, &d__2, &xr, &xi);
sr -= xr;
si -= xi;
goto L340;
/* .......... FORM EXCEPTIONAL SHIFT .......... */
L320:
sr = (d__1 = hr[en + enm1 * hr_dim1], abs(d__1)) + (d__2 = hr[enm1 + (en
- 2) * hr_dim1], abs(d__2));
si = 0.;
L340:
i__1 = en;
for (i__ = *low; i__ <= i__1; ++i__) {
hr[i__ + i__ * hr_dim1] -= sr;
hi[i__ + i__ * hi_dim1] -= si;
/* L360: */
}
tr += sr;
ti += si;
++its;
--itn;
/* .......... REDUCE TO TRIANGLE (ROWS) .......... */
lp1 = l + 1;
i__1 = en;
for (i__ = lp1; i__ <= i__1; ++i__) {
sr = hr[i__ + (i__ - 1) * hr_dim1];
hr[i__ + (i__ - 1) * hr_dim1] = 0.;
d__1 = pythag_(&hr[i__ - 1 + (i__ - 1) * hr_dim1], &hi[i__ - 1 + (i__
- 1) * hi_dim1]);
norm = pythag_(&d__1, &sr);
xr = hr[i__ - 1 + (i__ - 1) * hr_dim1] / norm;
wr[i__ - 1] = xr;
xi = hi[i__ - 1 + (i__ - 1) * hi_dim1] / norm;
wi[i__ - 1] = xi;
hr[i__ - 1 + (i__ - 1) * hr_dim1] = norm;
hi[i__ - 1 + (i__ - 1) * hi_dim1] = 0.;
hi[i__ + (i__ - 1) * hi_dim1] = sr / norm;
i__2 = en;
for (j = i__; j <= i__2; ++j) {
yr = hr[i__ - 1 + j * hr_dim1];
yi = hi[i__ - 1 + j * hi_dim1];
zzr = hr[i__ + j * hr_dim1];
zzi = hi[i__ + j * hi_dim1];
hr[i__ - 1 + j * hr_dim1] = xr * yr + xi * yi + hi[i__ + (i__ - 1)
* hi_dim1] * zzr;
hi[i__ - 1 + j * hi_dim1] = xr * yi - xi * yr + hi[i__ + (i__ - 1)
* hi_dim1] * zzi;
hr[i__ + j * hr_dim1] = xr * zzr - xi * zzi - hi[i__ + (i__ - 1) *
hi_dim1] * yr;
hi[i__ + j * hi_dim1] = xr * zzi + xi * zzr - hi[i__ + (i__ - 1) *
hi_dim1] * yi;
/* L490: */
}
/* L500: */
}
si = hi[en + en * hi_dim1];
if (si == 0.) {
goto L540;
}
norm = pythag_(&hr[en + en * hr_dim1], &si);
sr = hr[en + en * hr_dim1] / norm;
si /= norm;
hr[en + en * hr_dim1] = norm;
hi[en + en * hi_dim1] = 0.;
/* .......... INVERSE OPERATION (COLUMNS) .......... */
L540:
i__1 = en;
for (j = lp1; j <= i__1; ++j) {
xr = wr[j - 1];
xi = wi[j - 1];
i__2 = j;
for (i__ = l; i__ <= i__2; ++i__) {
yr = hr[i__ + (j - 1) * hr_dim1];
yi = 0.;
zzr = hr[i__ + j * hr_dim1];
zzi = hi[i__ + j * hi_dim1];
if (i__ == j) {
goto L560;
}
yi = hi[i__ + (j - 1) * hi_dim1];
hi[i__ + (j - 1) * hi_dim1] = xr * yi + xi * yr + hi[j + (j - 1) *
hi_dim1] * zzi;
L560:
hr[i__ + (j - 1) * hr_dim1] = xr * yr - xi * yi + hi[j + (j - 1) *
hi_dim1] * zzr;
hr[i__ + j * hr_dim1] = xr * zzr + xi * zzi - hi[j + (j - 1) *
hi_dim1] * yr;
hi[i__ + j * hi_dim1] = xr * zzi - xi * zzr - hi[j + (j - 1) *
hi_dim1] * yi;
/* L580: */
}
/* L600: */
}
if (si == 0.) {
goto L240;
}
i__1 = en;
for (i__ = l; i__ <= i__1; ++i__) {
yr = hr[i__ + en * hr_dim1];
yi = hi[i__ + en * hi_dim1];
hr[i__ + en * hr_dim1] = sr * yr - si * yi;
hi[i__ + en * hi_dim1] = sr * yi + si * yr;
/* L630: */
}
goto L240;
/* .......... A ROOT FOUND .......... */
L660:
wr[en] = hr[en + en * hr_dim1] + tr;
wi[en] = hi[en + en * hi_dim1] + ti;
en = enm1;
goto L220;
/* .......... SET ERROR -- ALL EIGENVALUES HAVE NOT */
/* CONVERGED AFTER 30*N ITERATIONS .......... */
L1000:
*ierr = en;
L1001:
return 0;
} /* comqr_ */
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