/* comlr2.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 comlr2_(integer *nm, integer *n, integer *low, integer * igh, integer *int__, doublereal *hr, doublereal *hi, doublereal *wr, doublereal *wi, doublereal *zr, doublereal *zi, integer *ierr) { /* System generated locals */ integer hr_dim1, hr_offset, hi_dim1, hi_offset, zr_dim1, zr_offset, zi_dim1, zi_offset, i__1, i__2, i__3; doublereal d__1, d__2, d__3, d__4; /* Local variables */ static integer iend; extern /* Subroutine */ int cdiv_(doublereal *, doublereal *, doublereal * , doublereal *, doublereal *, doublereal *); static doublereal norm; static integer i__, j, k, l, m, ii, en, jj, ll, mm, nn; static doublereal si, ti, xi, yi, sr, tr, xr, yr; static integer im1; extern /* Subroutine */ int csroot_(doublereal *, doublereal *, doublereal *, doublereal *); static integer ip1, mp1, itn, its; static doublereal zzi, zzr; static integer enm1; static doublereal tst1, tst2; /* THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE COMLR2, */ /* NUM. MATH. 16, 181-204(1970) BY PETERS AND WILKINSON. */ /* HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971). */ /* THIS SUBROUTINE FINDS THE EIGENVALUES AND EIGENVECTORS */ /* OF A COMPLEX UPPER HESSENBERG MATRIX BY THE MODIFIED LR */ /* METHOD. THE EIGENVECTORS OF A COMPLEX GENERAL MATRIX */ /* CAN ALSO BE FOUND IF COMHES HAS BEEN USED TO REDUCE */ /* THIS GENERAL MATRIX TO HESSENBERG FORM. */ /* 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. */ /* INT CONTAINS INFORMATION ON THE ROWS AND COLUMNS INTERCHANGED */ /* IN THE REDUCTION BY COMHES, IF PERFORMED. ONLY ELEMENTS */ /* LOW THROUGH IGH ARE USED. IF THE EIGENVECTORS OF THE HESSEN- */ /* BERG MATRIX ARE DESIRED, SET INT(J)=J FOR THESE ELEMENTS. */ /* HR AND HI CONTAIN THE REAL AND IMAGINARY PARTS, */ /* RESPECTIVELY, OF THE COMPLEX UPPER HESSENBERG MATRIX. */ /* THEIR LOWER TRIANGLES BELOW THE SUBDIAGONAL CONTAIN THE */ /* MULTIPLIERS WHICH WERE USED IN THE REDUCTION BY COMHES, */ /* IF PERFORMED. IF THE EIGENVECTORS OF THE HESSENBERG */ /* MATRIX ARE DESIRED, THESE ELEMENTS MUST BE SET TO ZERO. */ /* ON OUTPUT */ /* THE UPPER HESSENBERG PORTIONS OF HR AND HI HAVE BEEN */ /* DESTROYED, BUT THE LOCATION HR(1,1) CONTAINS THE NORM */ /* OF THE TRIANGULARIZED MATRIX. */ /* 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. */ /* ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS, */ /* RESPECTIVELY, OF THE EIGENVECTORS. THE EIGENVECTORS */ /* ARE UNNORMALIZED. IF AN ERROR EXIT IS MADE, NONE OF */ /* THE EIGENVECTORS HAS BEEN FOUND. */ /* 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. */ /* 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 */ zi_dim1 = *nm; zi_offset = zi_dim1 + 1; zi -= zi_offset; zr_dim1 = *nm; zr_offset = zr_dim1 + 1; zr -= zr_offset; --wi; --wr; hi_dim1 = *nm; hi_offset = hi_dim1 + 1; hi -= hi_offset; hr_dim1 = *nm; hr_offset = hr_dim1 + 1; hr -= hr_offset; --int__; /* Function Body */ *ierr = 0; /* .......... INITIALIZE EIGENVECTOR MATRIX .......... */ i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { i__2 = *n; for (j = 1; j <= i__2; ++j) { zr[i__ + j * zr_dim1] = 0.; zi[i__ + j * zi_dim1] = 0.; if (i__ == j) { zr[i__ + j * zr_dim1] = 1.; } /* L100: */ } } /* .......... FORM THE MATRIX OF ACCUMULATED TRANSFORMATIONS */ /* FROM THE INFORMATION LEFT BY COMHES .......... */ iend = *igh - *low - 1; if (iend <= 0) { goto L180; } /* .......... FOR I=IGH-1 STEP -1 UNTIL LOW+1 DO -- .......... */ i__2 = iend; for (ii = 1; ii <= i__2; ++ii) { i__ = *igh - ii; ip1 = i__ + 1; i__1 = *igh; for (k = ip1; k <= i__1; ++k) { zr[k + i__ * zr_dim1] = hr[k + (i__ - 1) * hr_dim1]; zi[k + i__ * zi_dim1] = hi[k + (i__ - 1) * hi_dim1]; /* L120: */ } j = int__[i__]; if (i__ == j) { goto L160; } i__1 = *igh; for (k = i__; k <= i__1; ++k) { zr[i__ + k * zr_dim1] = zr[j + k * zr_dim1]; zi[i__ + k * zi_dim1] = zi[j + k * zi_dim1]; zr[j + k * zr_dim1] = 0.; zi[j + k * zi_dim1] = 0.; /* L140: */ } zr[j + i__ * zr_dim1] = 1.; L160: ; } /* .......... STORE ROOTS ISOLATED BY CBAL .......... */ L180: i__2 = *n; for (i__ = 1; i__ <= i__2; ++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 L680; } its = 0; enm1 = en - 1; /* .......... LOOK FOR SINGLE SMALL SUB-DIAGONAL ELEMENT */ /* FOR L=EN STEP -1 UNTIL LOW DO -- .......... */ L240: i__2 = en; for (ll = *low; ll <= i__2; ++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)) + (d__2 = hi[l + (l - 1) * hi_dim1], abs(d__2)); 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] - hi[enm1 + en * hi_dim1] * hi[en + enm1 * hi_dim1]; xi = hr[enm1 + en * hr_dim1] * hi[en + enm1 * hi_dim1] + 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 = (d__1 = hi[en + enm1 * hi_dim1], abs(d__1)) + (d__2 = hi[enm1 + (en - 2) * hi_dim1], abs(d__2)); L340: i__2 = en; for (i__ = *low; i__ <= i__2; ++i__) { hr[i__ + i__ * hr_dim1] -= sr; hi[i__ + i__ * hi_dim1] -= si; /* L360: */ } tr += sr; ti += si; ++its; --itn; /* .......... LOOK FOR TWO CONSECUTIVE SMALL */ /* SUB-DIAGONAL ELEMENTS .......... */ xr = (d__1 = hr[enm1 + enm1 * hr_dim1], abs(d__1)) + (d__2 = hi[enm1 + enm1 * hi_dim1], abs(d__2)); yr = (d__1 = hr[en + enm1 * hr_dim1], abs(d__1)) + (d__2 = hi[en + enm1 * hi_dim1], abs(d__2)); zzr = (d__1 = hr[en + en * hr_dim1], abs(d__1)) + (d__2 = hi[en + en * hi_dim1], abs(d__2)); /* .......... FOR M=EN-1 STEP -1 UNTIL L DO -- .......... */ i__2 = enm1; for (mm = l; mm <= i__2; ++mm) { m = enm1 + l - mm; if (m == l) { goto L420; } yi = yr; yr = (d__1 = hr[m + (m - 1) * hr_dim1], abs(d__1)) + (d__2 = hi[m + ( m - 1) * hi_dim1], abs(d__2)); xi = zzr; zzr = xr; xr = (d__1 = hr[m - 1 + (m - 1) * hr_dim1], abs(d__1)) + (d__2 = hi[m - 1 + (m - 1) * hi_dim1], abs(d__2)); tst1 = zzr / yi * (zzr + xr + xi); tst2 = tst1 + yr; if (tst2 == tst1) { goto L420; } /* L380: */ } /* .......... TRIANGULAR DECOMPOSITION H=L*R .......... */ L420: mp1 = m + 1; i__2 = en; for (i__ = mp1; i__ <= i__2; ++i__) { im1 = i__ - 1; xr = hr[im1 + im1 * hr_dim1]; xi = hi[im1 + im1 * hi_dim1]; yr = hr[i__ + im1 * hr_dim1]; yi = hi[i__ + im1 * hi_dim1]; if (abs(xr) + abs(xi) >= abs(yr) + abs(yi)) { goto L460; } /* .......... INTERCHANGE ROWS OF HR AND HI .......... */ i__1 = *n; for (j = im1; j <= i__1; ++j) { zzr = hr[im1 + j * hr_dim1]; hr[im1 + j * hr_dim1] = hr[i__ + j * hr_dim1]; hr[i__ + j * hr_dim1] = zzr; zzi = hi[im1 + j * hi_dim1]; hi[im1 + j * hi_dim1] = hi[i__ + j * hi_dim1]; hi[i__ + j * hi_dim1] = zzi; /* L440: */ } cdiv_(&xr, &xi, &yr, &yi, &zzr, &zzi); wr[i__] = 1.; goto L480; L460: cdiv_(&yr, &yi, &xr, &xi, &zzr, &zzi); wr[i__] = -1.; L480: hr[i__ + im1 * hr_dim1] = zzr; hi[i__ + im1 * hi_dim1] = zzi; i__1 = *n; for (j = i__; j <= i__1; ++j) { hr[i__ + j * hr_dim1] = hr[i__ + j * hr_dim1] - zzr * hr[im1 + j * hr_dim1] + zzi * hi[im1 + j * hi_dim1]; hi[i__ + j * hi_dim1] = hi[i__ + j * hi_dim1] - zzr * hi[im1 + j * hi_dim1] - zzi * hr[im1 + j * hr_dim1]; /* L500: */ } /* L520: */ } /* .......... COMPOSITION R*L=H .......... */ i__2 = en; for (j = mp1; j <= i__2; ++j) { xr = hr[j + (j - 1) * hr_dim1]; xi = hi[j + (j - 1) * hi_dim1]; hr[j + (j - 1) * hr_dim1] = 0.; hi[j + (j - 1) * hi_dim1] = 0.; /* .......... INTERCHANGE COLUMNS OF HR, HI, ZR, AND ZI, */ /* IF NECESSARY .......... */ if (wr[j] <= 0.) { goto L580; } i__1 = j; for (i__ = 1; i__ <= i__1; ++i__) { zzr = hr[i__ + (j - 1) * hr_dim1]; hr[i__ + (j - 1) * hr_dim1] = hr[i__ + j * hr_dim1]; hr[i__ + j * hr_dim1] = zzr; zzi = hi[i__ + (j - 1) * hi_dim1]; hi[i__ + (j - 1) * hi_dim1] = hi[i__ + j * hi_dim1]; hi[i__ + j * hi_dim1] = zzi; /* L540: */ } i__1 = *igh; for (i__ = *low; i__ <= i__1; ++i__) { zzr = zr[i__ + (j - 1) * zr_dim1]; zr[i__ + (j - 1) * zr_dim1] = zr[i__ + j * zr_dim1]; zr[i__ + j * zr_dim1] = zzr; zzi = zi[i__ + (j - 1) * zi_dim1]; zi[i__ + (j - 1) * zi_dim1] = zi[i__ + j * zi_dim1]; zi[i__ + j * zi_dim1] = zzi; /* L560: */ } L580: i__1 = j; for (i__ = 1; i__ <= i__1; ++i__) { hr[i__ + (j - 1) * hr_dim1] = hr[i__ + (j - 1) * hr_dim1] + xr * hr[i__ + j * hr_dim1] - xi * hi[i__ + j * hi_dim1]; hi[i__ + (j - 1) * hi_dim1] = hi[i__ + (j - 1) * hi_dim1] + xr * hi[i__ + j * hi_dim1] + xi * hr[i__ + j * hr_dim1]; /* L600: */ } /* .......... ACCUMULATE TRANSFORMATIONS .......... */ i__1 = *igh; for (i__ = *low; i__ <= i__1; ++i__) { zr[i__ + (j - 1) * zr_dim1] = zr[i__ + (j - 1) * zr_dim1] + xr * zr[i__ + j * zr_dim1] - xi * zi[i__ + j * zi_dim1]; zi[i__ + (j - 1) * zi_dim1] = zi[i__ + (j - 1) * zi_dim1] + xr * zi[i__ + j * zi_dim1] + xi * zr[i__ + j * zr_dim1]; /* L620: */ } /* L640: */ } goto L240; /* .......... A ROOT FOUND .......... */ L660: hr[en + en * hr_dim1] += tr; wr[en] = hr[en + en * hr_dim1]; hi[en + en * hi_dim1] += ti; wi[en] = hi[en + en * hi_dim1]; en = enm1; goto L220; /* .......... ALL ROOTS FOUND. BACKSUBSTITUTE TO FIND */ /* VECTORS OF UPPER TRIANGULAR FORM .......... */ L680: norm = 0.; i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { i__1 = *n; for (j = i__; j <= i__1; ++j) { tr = (d__1 = hr[i__ + j * hr_dim1], abs(d__1)) + (d__2 = hi[i__ + j * hi_dim1], abs(d__2)); if (tr > norm) { norm = tr; } /* L720: */ } } hr[hr_dim1 + 1] = norm; if (*n == 1 || norm == 0.) { goto L1001; } /* .......... FOR EN=N STEP -1 UNTIL 2 DO -- .......... */ i__1 = *n; for (nn = 2; nn <= i__1; ++nn) { en = *n + 2 - nn; xr = wr[en]; xi = wi[en]; hr[en + en * hr_dim1] = 1.; hi[en + en * hi_dim1] = 0.; enm1 = en - 1; /* .......... FOR I=EN-1 STEP -1 UNTIL 1 DO -- .......... */ i__2 = enm1; for (ii = 1; ii <= i__2; ++ii) { i__ = en - ii; zzr = 0.; zzi = 0.; ip1 = i__ + 1; i__3 = en; for (j = ip1; j <= i__3; ++j) { zzr = zzr + hr[i__ + j * hr_dim1] * hr[j + en * hr_dim1] - hi[ i__ + j * hi_dim1] * hi[j + en * hi_dim1]; zzi = zzi + hr[i__ + j * hr_dim1] * hi[j + en * hi_dim1] + hi[ i__ + j * hi_dim1] * hr[j + en * hr_dim1]; /* L740: */ } yr = xr - wr[i__]; yi = xi - wi[i__]; if (yr != 0. || yi != 0.) { goto L765; } tst1 = norm; yr = tst1; L760: yr *= .01; tst2 = norm + yr; if (tst2 > tst1) { goto L760; } L765: cdiv_(&zzr, &zzi, &yr, &yi, &hr[i__ + en * hr_dim1], &hi[i__ + en * hi_dim1]); /* .......... OVERFLOW CONTROL .......... */ tr = (d__1 = hr[i__ + en * hr_dim1], abs(d__1)) + (d__2 = hi[i__ + en * hi_dim1], abs(d__2)); if (tr == 0.) { goto L780; } tst1 = tr; tst2 = tst1 + 1. / tst1; if (tst2 > tst1) { goto L780; } i__3 = en; for (j = i__; j <= i__3; ++j) { hr[j + en * hr_dim1] /= tr; hi[j + en * hi_dim1] /= tr; /* L770: */ } L780: ; } /* L800: */ } /* .......... END BACKSUBSTITUTION .......... */ enm1 = *n - 1; /* .......... VECTORS OF ISOLATED ROOTS .......... */ i__1 = enm1; for (i__ = 1; i__ <= i__1; ++i__) { if (i__ >= *low && i__ <= *igh) { goto L840; } ip1 = i__ + 1; i__2 = *n; for (j = ip1; j <= i__2; ++j) { zr[i__ + j * zr_dim1] = hr[i__ + j * hr_dim1]; zi[i__ + j * zi_dim1] = hi[i__ + j * hi_dim1]; /* L820: */ } L840: ; } /* .......... MULTIPLY BY TRANSFORMATION MATRIX TO GIVE */ /* VECTORS OF ORIGINAL FULL MATRIX. */ /* FOR J=N STEP -1 UNTIL LOW+1 DO -- .......... */ i__1 = enm1; for (jj = *low; jj <= i__1; ++jj) { j = *n + *low - jj; m = min(j,*igh); i__2 = *igh; for (i__ = *low; i__ <= i__2; ++i__) { zzr = 0.; zzi = 0.; i__3 = m; for (k = *low; k <= i__3; ++k) { zzr = zzr + zr[i__ + k * zr_dim1] * hr[k + j * hr_dim1] - zi[ i__ + k * zi_dim1] * hi[k + j * hi_dim1]; zzi = zzi + zr[i__ + k * zr_dim1] * hi[k + j * hi_dim1] + zi[ i__ + k * zi_dim1] * hr[k + j * hr_dim1]; /* L860: */ } zr[i__ + j * zr_dim1] = zzr; zi[i__ + j * zi_dim1] = zzi; /* L880: */ } } goto L1001; /* .......... SET ERROR -- ALL EIGENVALUES HAVE NOT */ /* CONVERGED AFTER 30*N ITERATIONS .......... */ L1000: *ierr = en; L1001: return 0; } /* comlr2_ */