#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int ssytd2_(char *uplo, integer *n, real *a, integer *lda, real *d__, real *e, real *tau, integer *info) { /* -- LAPACK routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= SSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q' * A * Q = T. Arguments ========= UPLO (input) CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular N (input) INTEGER The order of the matrix A. N >= 0. A (input/output) REAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). D (output) REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). E (output) REAL array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. TAU (output) REAL array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details). INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. Further Details =============== If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n-1) . . . H(2) H(1). Each H(i) has the form H(i) = I - tau * v * v' where tau is a real scalar, and v is a real vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i). If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(n-1). Each H(i) has the form H(i) = I - tau * v * v' where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i). The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). ===================================================================== Test the input parameters Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static real c_b8 = 0.f; static real c_b14 = -1.f; /* System generated locals */ integer a_dim1, a_offset, i__1, i__2, i__3; /* Local variables */ static integer i__; static real taui; extern doublereal sdot_(integer *, real *, integer *, real *, integer *); extern /* Subroutine */ int ssyr2_(char *, integer *, real *, real *, integer *, real *, integer *, real *, integer *); static real alpha; extern logical lsame_(char *, char *); static logical upper; extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *, real *, integer *), ssymv_(char *, integer *, real *, real *, integer *, real *, integer *, real *, real *, integer *), xerbla_(char *, integer *), slarfg_(integer *, real *, real *, integer *, real *); a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; --d__; --e; --tau; /* Function Body */ *info = 0; upper = lsame_(uplo, "U"); if (! upper && ! lsame_(uplo, "L")) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*lda < max(1,*n)) { *info = -4; } if (*info != 0) { i__1 = -(*info); xerbla_("SSYTD2", &i__1); return 0; } /* Quick return if possible */ if (*n <= 0) { return 0; } if (upper) { /* Reduce the upper triangle of A */ for (i__ = *n - 1; i__ >= 1; --i__) { /* Generate elementary reflector H(i) = I - tau * v * v' to annihilate A(1:i-1,i+1) */ slarfg_(&i__, &a[i__ + (i__ + 1) * a_dim1], &a[(i__ + 1) * a_dim1 + 1], &c__1, &taui); e[i__] = a[i__ + (i__ + 1) * a_dim1]; if (taui != 0.f) { /* Apply H(i) from both sides to A(1:i,1:i) */ a[i__ + (i__ + 1) * a_dim1] = 1.f; /* Compute x := tau * A * v storing x in TAU(1:i) */ ssymv_(uplo, &i__, &taui, &a[a_offset], lda, &a[(i__ + 1) * a_dim1 + 1], &c__1, &c_b8, &tau[1], &c__1); /* Compute w := x - 1/2 * tau * (x'*v) * v */ alpha = taui * -.5f * sdot_(&i__, &tau[1], &c__1, &a[(i__ + 1) * a_dim1 + 1], &c__1); saxpy_(&i__, &alpha, &a[(i__ + 1) * a_dim1 + 1], &c__1, &tau[ 1], &c__1); /* Apply the transformation as a rank-2 update: A := A - v * w' - w * v' */ ssyr2_(uplo, &i__, &c_b14, &a[(i__ + 1) * a_dim1 + 1], &c__1, &tau[1], &c__1, &a[a_offset], lda); a[i__ + (i__ + 1) * a_dim1] = e[i__]; } d__[i__ + 1] = a[i__ + 1 + (i__ + 1) * a_dim1]; tau[i__] = taui; /* L10: */ } d__[1] = a[a_dim1 + 1]; } else { /* Reduce the lower triangle of A */ i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { /* Generate elementary reflector H(i) = I - tau * v * v' to annihilate A(i+2:n,i) */ i__2 = *n - i__; /* Computing MIN */ i__3 = i__ + 2; slarfg_(&i__2, &a[i__ + 1 + i__ * a_dim1], &a[min(i__3,*n) + i__ * a_dim1], &c__1, &taui); e[i__] = a[i__ + 1 + i__ * a_dim1]; if (taui != 0.f) { /* Apply H(i) from both sides to A(i+1:n,i+1:n) */ a[i__ + 1 + i__ * a_dim1] = 1.f; /* Compute x := tau * A * v storing y in TAU(i:n-1) */ i__2 = *n - i__; ssymv_(uplo, &i__2, &taui, &a[i__ + 1 + (i__ + 1) * a_dim1], lda, &a[i__ + 1 + i__ * a_dim1], &c__1, &c_b8, &tau[ i__], &c__1); /* Compute w := x - 1/2 * tau * (x'*v) * v */ i__2 = *n - i__; alpha = taui * -.5f * sdot_(&i__2, &tau[i__], &c__1, &a[i__ + 1 + i__ * a_dim1], &c__1); i__2 = *n - i__; saxpy_(&i__2, &alpha, &a[i__ + 1 + i__ * a_dim1], &c__1, &tau[ i__], &c__1); /* Apply the transformation as a rank-2 update: A := A - v * w' - w * v' */ i__2 = *n - i__; ssyr2_(uplo, &i__2, &c_b14, &a[i__ + 1 + i__ * a_dim1], &c__1, &tau[i__], &c__1, &a[i__ + 1 + (i__ + 1) * a_dim1], lda); a[i__ + 1 + i__ * a_dim1] = e[i__]; } d__[i__] = a[i__ + i__ * a_dim1]; tau[i__] = taui; /* L20: */ } d__[*n] = a[*n + *n * a_dim1]; } return 0; /* End of SSYTD2 */ } /* ssytd2_ */