#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int sggbal_(char *job, integer *n, real *a, integer *lda, real *b, integer *ldb, integer *ilo, integer *ihi, real *lscale, real *rscale, real *work, integer *info) { /* -- LAPACK routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= SGGBAL balances a pair of general real matrices (A,B). This involves, first, permuting A and B by similarity transformations 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 matrices, and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized eigenvalue problem A*x = lambda*B*x. Arguments ========= JOB (input) CHARACTER*1 Specifies the operations to be performed on A and B: = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0 and RSCALE(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 matrices A and B. N >= 0. A (input/output) REAL 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. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). B (input/output) REAL array, dimension (LDB,N) On entry, the input matrix B. On exit, B is overwritten by the balanced matrix. If JOB = 'N', B is not referenced. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). ILO (output) INTEGER IHI (output) INTEGER ILO and IHI are set to integers such that on exit A(i,j) = 0 and B(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. LSCALE (output) REAL array, dimension (N) Details of the permutations and scaling factors applied to the left side of A and B. If P(j) is the index of the row interchanged with row j, and D(j) is the scaling factor applied to row j, then LSCALE(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. RSCALE (output) REAL array, dimension (N) Details of the permutations and scaling factors applied to the right side of A and B. If P(j) is the index of the column interchanged with column j, and D(j) is the scaling factor applied to column j, then LSCALE(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. WORK (workspace) REAL array, dimension (lwork) lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and at least 1 when JOB = 'N' or 'P'. INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. Further Details =============== See R.C. WARD, Balancing the generalized eigenvalue problem, SIAM J. Sci. Stat. Comp. 2 (1981), 141-152. ===================================================================== Test the input parameters Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static real c_b35 = 10.f; static real c_b71 = .5f; /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2, i__3; real r__1, r__2, r__3; /* Builtin functions */ double r_lg10(real *), r_sign(real *, real *), pow_ri(real *, integer *); /* Local variables */ static integer i__, j, k, l, m; static real t; static integer jc; static real ta, tb, tc; static integer ir; static real ew; static integer it, nr, ip1, jp1, lm1; static real cab, rab, ewc, cor, sum; static integer nrp2, icab, lcab; static real beta, coef; static integer irab, lrab; static real basl, cmax; extern doublereal sdot_(integer *, real *, integer *, real *, integer *); static real coef2, coef5, gamma, alpha; extern logical lsame_(char *, char *); extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *); static real sfmin, sfmax; static integer iflow; extern /* Subroutine */ int sswap_(integer *, real *, integer *, real *, integer *); static integer kount; extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *, real *, integer *); static real pgamma; extern doublereal slamch_(char *); extern /* Subroutine */ int xerbla_(char *, integer *); extern integer isamax_(integer *, real *, integer *); static integer lsfmin, lsfmax; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; --lscale; --rscale; --work; /* 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; } else if (*ldb < max(1,*n)) { *info = -6; } if (*info != 0) { i__1 = -(*info); xerbla_("SGGBAL", &i__1); return 0; } /* Quick return if possible */ if (*n == 0) { *ilo = 1; *ihi = *n; return 0; } if (*n == 1) { *ilo = 1; *ihi = *n; lscale[1] = 1.f; rscale[1] = 1.f; return 0; } if (lsame_(job, "N")) { *ilo = 1; *ihi = *n; i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { lscale[i__] = 1.f; rscale[i__] = 1.f; /* L10: */ } return 0; } k = 1; l = *n; if (lsame_(job, "S")) { goto L190; } goto L30; /* Permute the matrices A and B to isolate the eigenvalues. Find row with one nonzero in columns 1 through L */ L20: l = lm1; if (l != 1) { goto L30; } rscale[1] = 1.f; lscale[1] = 1.f; goto L190; L30: lm1 = l - 1; for (i__ = l; i__ >= 1; --i__) { i__1 = lm1; for (j = 1; j <= i__1; ++j) { jp1 = j + 1; if (a[i__ + j * a_dim1] != 0.f || b[i__ + j * b_dim1] != 0.f) { goto L50; } /* L40: */ } j = l; goto L70; L50: i__1 = l; for (j = jp1; j <= i__1; ++j) { if (a[i__ + j * a_dim1] != 0.f || b[i__ + j * b_dim1] != 0.f) { goto L80; } /* L60: */ } j = jp1 - 1; L70: m = l; iflow = 1; goto L160; L80: ; } goto L100; /* Find column with one nonzero in rows K through N */ L90: ++k; L100: i__1 = l; for (j = k; j <= i__1; ++j) { i__2 = lm1; for (i__ = k; i__ <= i__2; ++i__) { ip1 = i__ + 1; if (a[i__ + j * a_dim1] != 0.f || b[i__ + j * b_dim1] != 0.f) { goto L120; } /* L110: */ } i__ = l; goto L140; L120: i__2 = l; for (i__ = ip1; i__ <= i__2; ++i__) { if (a[i__ + j * a_dim1] != 0.f || b[i__ + j * b_dim1] != 0.f) { goto L150; } /* L130: */ } i__ = ip1 - 1; L140: m = k; iflow = 2; goto L160; L150: ; } goto L190; /* Permute rows M and I */ L160: lscale[m] = (real) i__; if (i__ == m) { goto L170; } i__1 = *n - k + 1; sswap_(&i__1, &a[i__ + k * a_dim1], lda, &a[m + k * a_dim1], lda); i__1 = *n - k + 1; sswap_(&i__1, &b[i__ + k * b_dim1], ldb, &b[m + k * b_dim1], ldb); /* Permute columns M and J */ L170: rscale[m] = (real) j; if (j == m) { goto L180; } sswap_(&l, &a[j * a_dim1 + 1], &c__1, &a[m * a_dim1 + 1], &c__1); sswap_(&l, &b[j * b_dim1 + 1], &c__1, &b[m * b_dim1 + 1], &c__1); L180: switch (iflow) { case 1: goto L20; case 2: goto L90; } L190: *ilo = k; *ihi = l; if (lsame_(job, "P")) { i__1 = *ihi; for (i__ = *ilo; i__ <= i__1; ++i__) { lscale[i__] = 1.f; rscale[i__] = 1.f; /* L195: */ } return 0; } if (*ilo == *ihi) { return 0; } /* Balance the submatrix in rows ILO to IHI. */ nr = *ihi - *ilo + 1; i__1 = *ihi; for (i__ = *ilo; i__ <= i__1; ++i__) { rscale[i__] = 0.f; lscale[i__] = 0.f; work[i__] = 0.f; work[i__ + *n] = 0.f; work[i__ + (*n << 1)] = 0.f; work[i__ + *n * 3] = 0.f; work[i__ + (*n << 2)] = 0.f; work[i__ + *n * 5] = 0.f; /* L200: */ } /* Compute right side vector in resulting linear equations */ basl = r_lg10(&c_b35); i__1 = *ihi; for (i__ = *ilo; i__ <= i__1; ++i__) { i__2 = *ihi; for (j = *ilo; j <= i__2; ++j) { tb = b[i__ + j * b_dim1]; ta = a[i__ + j * a_dim1]; if (ta == 0.f) { goto L210; } r__1 = dabs(ta); ta = r_lg10(&r__1) / basl; L210: if (tb == 0.f) { goto L220; } r__1 = dabs(tb); tb = r_lg10(&r__1) / basl; L220: work[i__ + (*n << 2)] = work[i__ + (*n << 2)] - ta - tb; work[j + *n * 5] = work[j + *n * 5] - ta - tb; /* L230: */ } /* L240: */ } coef = 1.f / (real) (nr << 1); coef2 = coef * coef; coef5 = coef2 * .5f; nrp2 = nr + 2; beta = 0.f; it = 1; /* Start generalized conjugate gradient iteration */ L250: gamma = sdot_(&nr, &work[*ilo + (*n << 2)], &c__1, &work[*ilo + (*n << 2)] , &c__1) + sdot_(&nr, &work[*ilo + *n * 5], &c__1, &work[*ilo + * n * 5], &c__1); ew = 0.f; ewc = 0.f; i__1 = *ihi; for (i__ = *ilo; i__ <= i__1; ++i__) { ew += work[i__ + (*n << 2)]; ewc += work[i__ + *n * 5]; /* L260: */ } /* Computing 2nd power */ r__1 = ew; /* Computing 2nd power */ r__2 = ewc; /* Computing 2nd power */ r__3 = ew - ewc; gamma = coef * gamma - coef2 * (r__1 * r__1 + r__2 * r__2) - coef5 * ( r__3 * r__3); if (gamma == 0.f) { goto L350; } if (it != 1) { beta = gamma / pgamma; } t = coef5 * (ewc - ew * 3.f); tc = coef5 * (ew - ewc * 3.f); sscal_(&nr, &beta, &work[*ilo], &c__1); sscal_(&nr, &beta, &work[*ilo + *n], &c__1); saxpy_(&nr, &coef, &work[*ilo + (*n << 2)], &c__1, &work[*ilo + *n], & c__1); saxpy_(&nr, &coef, &work[*ilo + *n * 5], &c__1, &work[*ilo], &c__1); i__1 = *ihi; for (i__ = *ilo; i__ <= i__1; ++i__) { work[i__] += tc; work[i__ + *n] += t; /* L270: */ } /* Apply matrix to vector */ i__1 = *ihi; for (i__ = *ilo; i__ <= i__1; ++i__) { kount = 0; sum = 0.f; i__2 = *ihi; for (j = *ilo; j <= i__2; ++j) { if (a[i__ + j * a_dim1] == 0.f) { goto L280; } ++kount; sum += work[j]; L280: if (b[i__ + j * b_dim1] == 0.f) { goto L290; } ++kount; sum += work[j]; L290: ; } work[i__ + (*n << 1)] = (real) kount * work[i__ + *n] + sum; /* L300: */ } i__1 = *ihi; for (j = *ilo; j <= i__1; ++j) { kount = 0; sum = 0.f; i__2 = *ihi; for (i__ = *ilo; i__ <= i__2; ++i__) { if (a[i__ + j * a_dim1] == 0.f) { goto L310; } ++kount; sum += work[i__ + *n]; L310: if (b[i__ + j * b_dim1] == 0.f) { goto L320; } ++kount; sum += work[i__ + *n]; L320: ; } work[j + *n * 3] = (real) kount * work[j] + sum; /* L330: */ } sum = sdot_(&nr, &work[*ilo + *n], &c__1, &work[*ilo + (*n << 1)], &c__1) + sdot_(&nr, &work[*ilo], &c__1, &work[*ilo + *n * 3], &c__1); alpha = gamma / sum; /* Determine correction to current iteration */ cmax = 0.f; i__1 = *ihi; for (i__ = *ilo; i__ <= i__1; ++i__) { cor = alpha * work[i__ + *n]; if (dabs(cor) > cmax) { cmax = dabs(cor); } lscale[i__] += cor; cor = alpha * work[i__]; if (dabs(cor) > cmax) { cmax = dabs(cor); } rscale[i__] += cor; /* L340: */ } if (cmax < .5f) { goto L350; } r__1 = -alpha; saxpy_(&nr, &r__1, &work[*ilo + (*n << 1)], &c__1, &work[*ilo + (*n << 2)] , &c__1); r__1 = -alpha; saxpy_(&nr, &r__1, &work[*ilo + *n * 3], &c__1, &work[*ilo + *n * 5], & c__1); pgamma = gamma; ++it; if (it <= nrp2) { goto L250; } /* End generalized conjugate gradient iteration */ L350: sfmin = slamch_("S"); sfmax = 1.f / sfmin; lsfmin = (integer) (r_lg10(&sfmin) / basl + 1.f); lsfmax = (integer) (r_lg10(&sfmax) / basl); i__1 = *ihi; for (i__ = *ilo; i__ <= i__1; ++i__) { i__2 = *n - *ilo + 1; irab = isamax_(&i__2, &a[i__ + *ilo * a_dim1], lda); rab = (r__1 = a[i__ + (irab + *ilo - 1) * a_dim1], dabs(r__1)); i__2 = *n - *ilo + 1; irab = isamax_(&i__2, &b[i__ + *ilo * b_dim1], ldb); /* Computing MAX */ r__2 = rab, r__3 = (r__1 = b[i__ + (irab + *ilo - 1) * b_dim1], dabs( r__1)); rab = dmax(r__2,r__3); r__1 = rab + sfmin; lrab = (integer) (r_lg10(&r__1) / basl + 1.f); ir = lscale[i__] + r_sign(&c_b71, &lscale[i__]); /* Computing MIN */ i__2 = max(ir,lsfmin), i__2 = min(i__2,lsfmax), i__3 = lsfmax - lrab; ir = min(i__2,i__3); lscale[i__] = pow_ri(&c_b35, &ir); icab = isamax_(ihi, &a[i__ * a_dim1 + 1], &c__1); cab = (r__1 = a[icab + i__ * a_dim1], dabs(r__1)); icab = isamax_(ihi, &b[i__ * b_dim1 + 1], &c__1); /* Computing MAX */ r__2 = cab, r__3 = (r__1 = b[icab + i__ * b_dim1], dabs(r__1)); cab = dmax(r__2,r__3); r__1 = cab + sfmin; lcab = (integer) (r_lg10(&r__1) / basl + 1.f); jc = rscale[i__] + r_sign(&c_b71, &rscale[i__]); /* Computing MIN */ i__2 = max(jc,lsfmin), i__2 = min(i__2,lsfmax), i__3 = lsfmax - lcab; jc = min(i__2,i__3); rscale[i__] = pow_ri(&c_b35, &jc); /* L360: */ } /* Row scaling of matrices A and B */ i__1 = *ihi; for (i__ = *ilo; i__ <= i__1; ++i__) { i__2 = *n - *ilo + 1; sscal_(&i__2, &lscale[i__], &a[i__ + *ilo * a_dim1], lda); i__2 = *n - *ilo + 1; sscal_(&i__2, &lscale[i__], &b[i__ + *ilo * b_dim1], ldb); /* L370: */ } /* Column scaling of matrices A and B */ i__1 = *ihi; for (j = *ilo; j <= i__1; ++j) { sscal_(ihi, &rscale[j], &a[j * a_dim1 + 1], &c__1); sscal_(ihi, &rscale[j], &b[j * b_dim1 + 1], &c__1); /* L380: */ } return 0; /* End of SGGBAL */ } /* sggbal_ */