#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int dgebal_(char *job, integer *n, doublereal *a, integer * lda, integer *ilo, integer *ihi, doublereal *scale, integer *info) { /* -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University June 30, 1999 Purpose ======= DGEBAL balances a general real matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors. Arguments ========= JOB (input) CHARACTER*1 Specifies the operations to be performed on A: = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0 for i = 1,...,N; = 'P': permute only; = 'S': scale only; = 'B': both permute and scale. N (input) INTEGER The order of the matrix A. N >= 0. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the input matrix A. On exit, A is overwritten by the balanced matrix. If JOB = 'N', A is not referenced. See Further Details. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). ILO (output) INTEGER IHI (output) INTEGER ILO and IHI are set to integers such that on exit A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1 and IHI = N. SCALE (output) DOUBLE PRECISION array, dimension (N) Details of the permutations and scaling factors applied to A. If P(j) is the index of the row and column interchanged with row and column j and D(j) is the scaling factor applied to row and column j, then SCALE(j) = P(j) for j = 1,...,ILO-1 = D(j) for j = ILO,...,IHI = P(j) for j = IHI+1,...,N. The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. INFO (output) INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. Further Details =============== The permutations consist of row and column interchanges which put the matrix in the form ( T1 X Y ) P A P = ( 0 B Z ) ( 0 0 T2 ) where T1 and T2 are upper triangular matrices whose eigenvalues lie along the diagonal. The column indices ILO and IHI mark the starting and ending columns of the submatrix B. Balancing consists of applying a diagonal similarity transformation inv(D) * B * D to make the 1-norms of each row of B and its corresponding column nearly equal. The output matrix is ( T1 X*D Y ) ( 0 inv(D)*B*D inv(D)*Z ). ( 0 0 T2 ) Information about the permutations P and the diagonal matrix D is returned in the vector SCALE. This subroutine is based on the EISPACK routine BALANC. Modified by Tzu-Yi Chen, Computer Science Division, University of California at Berkeley, USA ===================================================================== Test the input parameters Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; /* System generated locals */ integer a_dim1, a_offset, i__1, i__2; doublereal d__1, d__2; /* Local variables */ static integer iexc; static doublereal c__, f, g; static integer i__, j, k, l, m; static doublereal r__, s; extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, integer *); extern logical lsame_(char *, char *); extern /* Subroutine */ int dswap_(integer *, doublereal *, integer *, doublereal *, integer *); static doublereal sfmin1, sfmin2, sfmax1, sfmax2, ca, ra; extern doublereal dlamch_(char *); extern integer idamax_(integer *, doublereal *, integer *); extern /* Subroutine */ int xerbla_(char *, integer *); static logical noconv; static integer ica, ira; #define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1] a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; --scale; /* Function Body */ *info = 0; if (! lsame_(job, "N") && ! lsame_(job, "P") && ! lsame_(job, "S") && ! lsame_(job, "B")) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*lda < max(1,*n)) { *info = -4; } if (*info != 0) { i__1 = -(*info); xerbla_("DGEBAL", &i__1); return 0; } k = 1; l = *n; if (*n == 0) { goto L210; } if (lsame_(job, "N")) { i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { scale[i__] = 1.; /* L10: */ } goto L210; } if (lsame_(job, "S")) { goto L120; } /* Permutation to isolate eigenvalues if possible */ goto L50; /* Row and column exchange. */ L20: scale[m] = (doublereal) j; if (j == m) { goto L30; } dswap_(&l, &a_ref(1, j), &c__1, &a_ref(1, m), &c__1); i__1 = *n - k + 1; dswap_(&i__1, &a_ref(j, k), lda, &a_ref(m, k), lda); L30: switch (iexc) { case 1: goto L40; case 2: goto L80; } /* Search for rows isolating an eigenvalue and push them down. */ L40: if (l == 1) { goto L210; } --l; L50: for (j = l; j >= 1; --j) { i__1 = l; for (i__ = 1; i__ <= i__1; ++i__) { if (i__ == j) { goto L60; } if (a_ref(j, i__) != 0.) { goto L70; } L60: ; } m = l; iexc = 1; goto L20; L70: ; } goto L90; /* Search for columns isolating an eigenvalue and push them left. */ L80: ++k; L90: i__1 = l; for (j = k; j <= i__1; ++j) { i__2 = l; for (i__ = k; i__ <= i__2; ++i__) { if (i__ == j) { goto L100; } if (a_ref(i__, j) != 0.) { goto L110; } L100: ; } m = k; iexc = 2; goto L20; L110: ; } L120: i__1 = l; for (i__ = k; i__ <= i__1; ++i__) { scale[i__] = 1.; /* L130: */ } if (lsame_(job, "P")) { goto L210; } /* Balance the submatrix in rows K to L. Iterative loop for norm reduction */ sfmin1 = dlamch_("S") / dlamch_("P"); sfmax1 = 1. / sfmin1; sfmin2 = sfmin1 * 8.; sfmax2 = 1. / sfmin2; L140: noconv = FALSE_; i__1 = l; for (i__ = k; i__ <= i__1; ++i__) { c__ = 0.; r__ = 0.; i__2 = l; for (j = k; j <= i__2; ++j) { if (j == i__) { goto L150; } c__ += (d__1 = a_ref(j, i__), abs(d__1)); r__ += (d__1 = a_ref(i__, j), abs(d__1)); L150: ; } ica = idamax_(&l, &a_ref(1, i__), &c__1); ca = (d__1 = a_ref(ica, i__), abs(d__1)); i__2 = *n - k + 1; ira = idamax_(&i__2, &a_ref(i__, k), lda); ra = (d__1 = a_ref(i__, ira + k - 1), abs(d__1)); /* Guard against zero C or R due to underflow. */ if (c__ == 0. || r__ == 0.) { goto L200; } g = r__ / 8.; f = 1.; s = c__ + r__; L160: /* Computing MAX */ d__1 = max(f,c__); /* Computing MIN */ d__2 = min(r__,g); if (c__ >= g || max(d__1,ca) >= sfmax2 || min(d__2,ra) <= sfmin2) { goto L170; } f *= 8.; c__ *= 8.; ca *= 8.; r__ /= 8.; g /= 8.; ra /= 8.; goto L160; L170: g = c__ / 8.; L180: /* Computing MIN */ d__1 = min(f,c__), d__1 = min(d__1,g); if (g < r__ || max(r__,ra) >= sfmax2 || min(d__1,ca) <= sfmin2) { goto L190; } f /= 8.; c__ /= 8.; g /= 8.; ca /= 8.; r__ *= 8.; ra *= 8.; goto L180; /* Now balance. */ L190: if (c__ + r__ >= s * .95) { goto L200; } if (f < 1. && scale[i__] < 1.) { if (f * scale[i__] <= sfmin1) { goto L200; } } if (f > 1. && scale[i__] > 1.) { if (scale[i__] >= sfmax1 / f) { goto L200; } } g = 1. / f; scale[i__] *= f; noconv = TRUE_; i__2 = *n - k + 1; dscal_(&i__2, &g, &a_ref(i__, k), lda); dscal_(&l, &f, &a_ref(1, i__), &c__1); L200: ; } if (noconv) { goto L140; } L210: *ilo = k; *ihi = l; return 0; /* End of DGEBAL */ } /* dgebal_ */ #undef a_ref