#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int chbtrd_(char *vect, char *uplo, integer *n, integer *kd, complex *ab, integer *ldab, real *d__, real *e, complex *q, integer * ldq, complex *work, integer *info) { /* -- LAPACK routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= CHBTRD reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. Arguments ========= VECT (input) CHARACTER*1 = 'N': do not form Q; = 'V': form Q; = 'U': update a matrix X, by forming X*Q. UPLO (input) CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N (input) INTEGER The order of the matrix A. N >= 0. KD (input) INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. AB (input/output) COMPLEX array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, the diagonal elements of AB are overwritten by the diagonal elements of the tridiagonal matrix T; if KD > 0, the elements on the first superdiagonal (if UPLO = 'U') or the first subdiagonal (if UPLO = 'L') are overwritten by the off-diagonal elements of T; the rest of AB is overwritten by values generated during the reduction. LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= KD+1. D (output) REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T. E (output) REAL array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. Q (input/output) COMPLEX array, dimension (LDQ,N) On entry, if VECT = 'U', then Q must contain an N-by-N matrix X; if VECT = 'N' or 'V', then Q need not be set. On exit: if VECT = 'V', Q contains the N-by-N unitary matrix Q; if VECT = 'U', Q contains the product X*Q; if VECT = 'N', the array Q is not referenced. LDQ (input) INTEGER The leading dimension of the array Q. LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. WORK (workspace) COMPLEX array, dimension (N) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value Further Details =============== Modified by Linda Kaufman, Bell Labs. ===================================================================== Test the input parameters Parameter adjustments */ /* Table of constant values */ static complex c_b1 = {0.f,0.f}; static complex c_b2 = {1.f,0.f}; static integer c__1 = 1; /* System generated locals */ integer ab_dim1, ab_offset, q_dim1, q_offset, i__1, i__2, i__3, i__4, i__5, i__6; real r__1; complex q__1; /* Builtin functions */ void r_cnjg(complex *, complex *); double c_abs(complex *); /* Local variables */ static integer i__, j, k, l; static complex t; static integer i2, j1, j2, nq, nr, kd1, ibl, iqb, kdn, jin, nrt, kdm1, inca, jend, lend, jinc; static real abst; static integer incx, last; static complex temp; extern /* Subroutine */ int crot_(integer *, complex *, integer *, complex *, integer *, real *, complex *); static integer j1end, j1inc; extern /* Subroutine */ int cscal_(integer *, complex *, complex *, integer *); static integer iqend; extern logical lsame_(char *, char *); static logical initq, wantq, upper; extern /* Subroutine */ int clar2v_(integer *, complex *, complex *, complex *, integer *, real *, complex *, integer *), clacgv_( integer *, complex *, integer *); static integer iqaend; extern /* Subroutine */ int claset_(char *, integer *, integer *, complex *, complex *, complex *, integer *), clartg_(complex *, complex *, real *, complex *, complex *), xerbla_(char *, integer *), clargv_(integer *, complex *, integer *, complex *, integer *, real *, integer *), clartv_(integer *, complex *, integer *, complex *, integer *, real *, complex *, integer *); ab_dim1 = *ldab; ab_offset = 1 + ab_dim1; ab -= ab_offset; --d__; --e; q_dim1 = *ldq; q_offset = 1 + q_dim1; q -= q_offset; --work; /* Function Body */ initq = lsame_(vect, "V"); wantq = initq || lsame_(vect, "U"); upper = lsame_(uplo, "U"); kd1 = *kd + 1; kdm1 = *kd - 1; incx = *ldab - 1; iqend = 1; *info = 0; if (! wantq && ! lsame_(vect, "N")) { *info = -1; } else if (! upper && ! lsame_(uplo, "L")) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*kd < 0) { *info = -4; } else if (*ldab < kd1) { *info = -6; } else if (*ldq < max(1,*n) && wantq) { *info = -10; } if (*info != 0) { i__1 = -(*info); xerbla_("CHBTRD", &i__1); return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } /* Initialize Q to the unit matrix, if needed */ if (initq) { claset_("Full", n, n, &c_b1, &c_b2, &q[q_offset], ldq); } /* Wherever possible, plane rotations are generated and applied in vector operations of length NR over the index set J1:J2:KD1. The real cosines and complex sines of the plane rotations are stored in the arrays D and WORK. */ inca = kd1 * *ldab; /* Computing MIN */ i__1 = *n - 1; kdn = min(i__1,*kd); if (upper) { if (*kd > 1) { /* Reduce to complex Hermitian tridiagonal form, working with the upper triangle */ nr = 0; j1 = kdn + 2; j2 = 1; i__1 = kd1 + ab_dim1; i__2 = kd1 + ab_dim1; r__1 = ab[i__2].r; ab[i__1].r = r__1, ab[i__1].i = 0.f; i__1 = *n - 2; for (i__ = 1; i__ <= i__1; ++i__) { /* Reduce i-th row of matrix to tridiagonal form */ for (k = kdn + 1; k >= 2; --k) { j1 += kdn; j2 += kdn; if (nr > 0) { /* generate plane rotations to annihilate nonzero elements which have been created outside the band */ clargv_(&nr, &ab[(j1 - 1) * ab_dim1 + 1], &inca, & work[j1], &kd1, &d__[j1], &kd1); /* apply rotations from the right Dependent on the the number of diagonals either CLARTV or CROT is used */ if (nr >= (*kd << 1) - 1) { i__2 = *kd - 1; for (l = 1; l <= i__2; ++l) { clartv_(&nr, &ab[l + 1 + (j1 - 1) * ab_dim1], &inca, &ab[l + j1 * ab_dim1], &inca, & d__[j1], &work[j1], &kd1); /* L10: */ } } else { jend = j1 + (nr - 1) * kd1; i__2 = jend; i__3 = kd1; for (jinc = j1; i__3 < 0 ? jinc >= i__2 : jinc <= i__2; jinc += i__3) { crot_(&kdm1, &ab[(jinc - 1) * ab_dim1 + 2], & c__1, &ab[jinc * ab_dim1 + 1], &c__1, &d__[jinc], &work[jinc]); /* L20: */ } } } if (k > 2) { if (k <= *n - i__ + 1) { /* generate plane rotation to annihilate a(i,i+k-1) within the band */ clartg_(&ab[*kd - k + 3 + (i__ + k - 2) * ab_dim1] , &ab[*kd - k + 2 + (i__ + k - 1) * ab_dim1], &d__[i__ + k - 1], &work[i__ + k - 1], &temp); i__3 = *kd - k + 3 + (i__ + k - 2) * ab_dim1; ab[i__3].r = temp.r, ab[i__3].i = temp.i; /* apply rotation from the right */ i__3 = k - 3; crot_(&i__3, &ab[*kd - k + 4 + (i__ + k - 2) * ab_dim1], &c__1, &ab[*kd - k + 3 + (i__ + k - 1) * ab_dim1], &c__1, &d__[i__ + k - 1], &work[i__ + k - 1]); } ++nr; j1 = j1 - kdn - 1; } /* apply plane rotations from both sides to diagonal blocks */ if (nr > 0) { clar2v_(&nr, &ab[kd1 + (j1 - 1) * ab_dim1], &ab[kd1 + j1 * ab_dim1], &ab[*kd + j1 * ab_dim1], &inca, &d__[j1], &work[j1], &kd1); } /* apply plane rotations from the left */ if (nr > 0) { clacgv_(&nr, &work[j1], &kd1); if ((*kd << 1) - 1 < nr) { /* Dependent on the the number of diagonals either CLARTV or CROT is used */ i__3 = *kd - 1; for (l = 1; l <= i__3; ++l) { if (j2 + l > *n) { nrt = nr - 1; } else { nrt = nr; } if (nrt > 0) { clartv_(&nrt, &ab[*kd - l + (j1 + l) * ab_dim1], &inca, &ab[*kd - l + 1 + (j1 + l) * ab_dim1], &inca, & d__[j1], &work[j1], &kd1); } /* L30: */ } } else { j1end = j1 + kd1 * (nr - 2); if (j1end >= j1) { i__3 = j1end; i__2 = kd1; for (jin = j1; i__2 < 0 ? jin >= i__3 : jin <= i__3; jin += i__2) { i__4 = *kd - 1; crot_(&i__4, &ab[*kd - 1 + (jin + 1) * ab_dim1], &incx, &ab[*kd + (jin + 1) * ab_dim1], &incx, &d__[jin], & work[jin]); /* L40: */ } } /* Computing MIN */ i__2 = kdm1, i__3 = *n - j2; lend = min(i__2,i__3); last = j1end + kd1; if (lend > 0) { crot_(&lend, &ab[*kd - 1 + (last + 1) * ab_dim1], &incx, &ab[*kd + (last + 1) * ab_dim1], &incx, &d__[last], &work[ last]); } } } if (wantq) { /* accumulate product of plane rotations in Q */ if (initq) { /* take advantage of the fact that Q was initially the Identity matrix */ iqend = max(iqend,j2); /* Computing MAX */ i__2 = 0, i__3 = k - 3; i2 = max(i__2,i__3); iqaend = i__ * *kd + 1; if (k == 2) { iqaend += *kd; } iqaend = min(iqaend,iqend); i__2 = j2; i__3 = kd1; for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j += i__3) { ibl = i__ - i2 / kdm1; ++i2; /* Computing MAX */ i__4 = 1, i__5 = j - ibl; iqb = max(i__4,i__5); nq = iqaend + 1 - iqb; /* Computing MIN */ i__4 = iqaend + *kd; iqaend = min(i__4,iqend); r_cnjg(&q__1, &work[j]); crot_(&nq, &q[iqb + (j - 1) * q_dim1], &c__1, &q[iqb + j * q_dim1], &c__1, &d__[j], &q__1); /* L50: */ } } else { i__3 = j2; i__2 = kd1; for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j += i__2) { r_cnjg(&q__1, &work[j]); crot_(n, &q[(j - 1) * q_dim1 + 1], &c__1, &q[ j * q_dim1 + 1], &c__1, &d__[j], & q__1); /* L60: */ } } } if (j2 + kdn > *n) { /* adjust J2 to keep within the bounds of the matrix */ --nr; j2 = j2 - kdn - 1; } i__2 = j2; i__3 = kd1; for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j += i__3) { /* create nonzero element a(j-1,j+kd) outside the band and store it in WORK */ i__4 = j + *kd; i__5 = j; i__6 = (j + *kd) * ab_dim1 + 1; q__1.r = work[i__5].r * ab[i__6].r - work[i__5].i * ab[i__6].i, q__1.i = work[i__5].r * ab[i__6] .i + work[i__5].i * ab[i__6].r; work[i__4].r = q__1.r, work[i__4].i = q__1.i; i__4 = (j + *kd) * ab_dim1 + 1; i__5 = j; i__6 = (j + *kd) * ab_dim1 + 1; q__1.r = d__[i__5] * ab[i__6].r, q__1.i = d__[i__5] * ab[i__6].i; ab[i__4].r = q__1.r, ab[i__4].i = q__1.i; /* L70: */ } /* L80: */ } /* L90: */ } } if (*kd > 0) { /* make off-diagonal elements real and copy them to E */ i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { i__3 = *kd + (i__ + 1) * ab_dim1; t.r = ab[i__3].r, t.i = ab[i__3].i; abst = c_abs(&t); i__3 = *kd + (i__ + 1) * ab_dim1; ab[i__3].r = abst, ab[i__3].i = 0.f; e[i__] = abst; if (abst != 0.f) { q__1.r = t.r / abst, q__1.i = t.i / abst; t.r = q__1.r, t.i = q__1.i; } else { t.r = 1.f, t.i = 0.f; } if (i__ < *n - 1) { i__3 = *kd + (i__ + 2) * ab_dim1; i__2 = *kd + (i__ + 2) * ab_dim1; q__1.r = ab[i__2].r * t.r - ab[i__2].i * t.i, q__1.i = ab[ i__2].r * t.i + ab[i__2].i * t.r; ab[i__3].r = q__1.r, ab[i__3].i = q__1.i; } if (wantq) { r_cnjg(&q__1, &t); cscal_(n, &q__1, &q[(i__ + 1) * q_dim1 + 1], &c__1); } /* L100: */ } } else { /* set E to zero if original matrix was diagonal */ i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { e[i__] = 0.f; /* L110: */ } } /* copy diagonal elements to D */ i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { i__3 = i__; i__2 = kd1 + i__ * ab_dim1; d__[i__3] = ab[i__2].r; /* L120: */ } } else { if (*kd > 1) { /* Reduce to complex Hermitian tridiagonal form, working with the lower triangle */ nr = 0; j1 = kdn + 2; j2 = 1; i__1 = ab_dim1 + 1; i__3 = ab_dim1 + 1; r__1 = ab[i__3].r; ab[i__1].r = r__1, ab[i__1].i = 0.f; i__1 = *n - 2; for (i__ = 1; i__ <= i__1; ++i__) { /* Reduce i-th column of matrix to tridiagonal form */ for (k = kdn + 1; k >= 2; --k) { j1 += kdn; j2 += kdn; if (nr > 0) { /* generate plane rotations to annihilate nonzero elements which have been created outside the band */ clargv_(&nr, &ab[kd1 + (j1 - kd1) * ab_dim1], &inca, & work[j1], &kd1, &d__[j1], &kd1); /* apply plane rotations from one side Dependent on the the number of diagonals either CLARTV or CROT is used */ if (nr > (*kd << 1) - 1) { i__3 = *kd - 1; for (l = 1; l <= i__3; ++l) { clartv_(&nr, &ab[kd1 - l + (j1 - kd1 + l) * ab_dim1], &inca, &ab[kd1 - l + 1 + ( j1 - kd1 + l) * ab_dim1], &inca, &d__[ j1], &work[j1], &kd1); /* L130: */ } } else { jend = j1 + kd1 * (nr - 1); i__3 = jend; i__2 = kd1; for (jinc = j1; i__2 < 0 ? jinc >= i__3 : jinc <= i__3; jinc += i__2) { crot_(&kdm1, &ab[*kd + (jinc - *kd) * ab_dim1] , &incx, &ab[kd1 + (jinc - *kd) * ab_dim1], &incx, &d__[jinc], &work[ jinc]); /* L140: */ } } } if (k > 2) { if (k <= *n - i__ + 1) { /* generate plane rotation to annihilate a(i+k-1,i) within the band */ clartg_(&ab[k - 1 + i__ * ab_dim1], &ab[k + i__ * ab_dim1], &d__[i__ + k - 1], &work[i__ + k - 1], &temp); i__2 = k - 1 + i__ * ab_dim1; ab[i__2].r = temp.r, ab[i__2].i = temp.i; /* apply rotation from the left */ i__2 = k - 3; i__3 = *ldab - 1; i__4 = *ldab - 1; crot_(&i__2, &ab[k - 2 + (i__ + 1) * ab_dim1], & i__3, &ab[k - 1 + (i__ + 1) * ab_dim1], & i__4, &d__[i__ + k - 1], &work[i__ + k - 1]); } ++nr; j1 = j1 - kdn - 1; } /* apply plane rotations from both sides to diagonal blocks */ if (nr > 0) { clar2v_(&nr, &ab[(j1 - 1) * ab_dim1 + 1], &ab[j1 * ab_dim1 + 1], &ab[(j1 - 1) * ab_dim1 + 2], & inca, &d__[j1], &work[j1], &kd1); } /* apply plane rotations from the right Dependent on the the number of diagonals either CLARTV or CROT is used */ if (nr > 0) { clacgv_(&nr, &work[j1], &kd1); if (nr > (*kd << 1) - 1) { i__2 = *kd - 1; for (l = 1; l <= i__2; ++l) { if (j2 + l > *n) { nrt = nr - 1; } else { nrt = nr; } if (nrt > 0) { clartv_(&nrt, &ab[l + 2 + (j1 - 1) * ab_dim1], &inca, &ab[l + 1 + j1 * ab_dim1], &inca, &d__[j1], &work[ j1], &kd1); } /* L150: */ } } else { j1end = j1 + kd1 * (nr - 2); if (j1end >= j1) { i__2 = j1end; i__3 = kd1; for (j1inc = j1; i__3 < 0 ? j1inc >= i__2 : j1inc <= i__2; j1inc += i__3) { crot_(&kdm1, &ab[(j1inc - 1) * ab_dim1 + 3], &c__1, &ab[j1inc * ab_dim1 + 2], &c__1, &d__[j1inc], &work[ j1inc]); /* L160: */ } } /* Computing MIN */ i__3 = kdm1, i__2 = *n - j2; lend = min(i__3,i__2); last = j1end + kd1; if (lend > 0) { crot_(&lend, &ab[(last - 1) * ab_dim1 + 3], & c__1, &ab[last * ab_dim1 + 2], &c__1, &d__[last], &work[last]); } } } if (wantq) { /* accumulate product of plane rotations in Q */ if (initq) { /* take advantage of the fact that Q was initially the Identity matrix */ iqend = max(iqend,j2); /* Computing MAX */ i__3 = 0, i__2 = k - 3; i2 = max(i__3,i__2); iqaend = i__ * *kd + 1; if (k == 2) { iqaend += *kd; } iqaend = min(iqaend,iqend); i__3 = j2; i__2 = kd1; for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j += i__2) { ibl = i__ - i2 / kdm1; ++i2; /* Computing MAX */ i__4 = 1, i__5 = j - ibl; iqb = max(i__4,i__5); nq = iqaend + 1 - iqb; /* Computing MIN */ i__4 = iqaend + *kd; iqaend = min(i__4,iqend); crot_(&nq, &q[iqb + (j - 1) * q_dim1], &c__1, &q[iqb + j * q_dim1], &c__1, &d__[j], &work[j]); /* L170: */ } } else { i__2 = j2; i__3 = kd1; for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j += i__3) { crot_(n, &q[(j - 1) * q_dim1 + 1], &c__1, &q[ j * q_dim1 + 1], &c__1, &d__[j], & work[j]); /* L180: */ } } } if (j2 + kdn > *n) { /* adjust J2 to keep within the bounds of the matrix */ --nr; j2 = j2 - kdn - 1; } i__3 = j2; i__2 = kd1; for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j += i__2) { /* create nonzero element a(j+kd,j-1) outside the band and store it in WORK */ i__4 = j + *kd; i__5 = j; i__6 = kd1 + j * ab_dim1; q__1.r = work[i__5].r * ab[i__6].r - work[i__5].i * ab[i__6].i, q__1.i = work[i__5].r * ab[i__6] .i + work[i__5].i * ab[i__6].r; work[i__4].r = q__1.r, work[i__4].i = q__1.i; i__4 = kd1 + j * ab_dim1; i__5 = j; i__6 = kd1 + j * ab_dim1; q__1.r = d__[i__5] * ab[i__6].r, q__1.i = d__[i__5] * ab[i__6].i; ab[i__4].r = q__1.r, ab[i__4].i = q__1.i; /* L190: */ } /* L200: */ } /* L210: */ } } if (*kd > 0) { /* make off-diagonal elements real and copy them to E */ i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { i__2 = i__ * ab_dim1 + 2; t.r = ab[i__2].r, t.i = ab[i__2].i; abst = c_abs(&t); i__2 = i__ * ab_dim1 + 2; ab[i__2].r = abst, ab[i__2].i = 0.f; e[i__] = abst; if (abst != 0.f) { q__1.r = t.r / abst, q__1.i = t.i / abst; t.r = q__1.r, t.i = q__1.i; } else { t.r = 1.f, t.i = 0.f; } if (i__ < *n - 1) { i__2 = (i__ + 1) * ab_dim1 + 2; i__3 = (i__ + 1) * ab_dim1 + 2; q__1.r = ab[i__3].r * t.r - ab[i__3].i * t.i, q__1.i = ab[ i__3].r * t.i + ab[i__3].i * t.r; ab[i__2].r = q__1.r, ab[i__2].i = q__1.i; } if (wantq) { cscal_(n, &t, &q[(i__ + 1) * q_dim1 + 1], &c__1); } /* L220: */ } } else { /* set E to zero if original matrix was diagonal */ i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { e[i__] = 0.f; /* L230: */ } } /* copy diagonal elements to D */ i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { i__2 = i__; i__3 = i__ * ab_dim1 + 1; d__[i__2] = ab[i__3].r; /* L240: */ } } return 0; /* End of CHBTRD */ } /* chbtrd_ */