#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int dsptrd_(char *uplo, integer *n, doublereal *ap, doublereal *d__, doublereal *e, doublereal *tau, integer *info) { /* -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University March 31, 1993 Purpose ======= DSPTRD reduces a real symmetric matrix A stored in packed form to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T. Arguments ========= 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. AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. 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. D (output) DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). E (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 AP, overwriting A(1:i-1,i+1), and tau is stored 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 AP, overwriting A(i+2:n,i), and tau is stored in TAU(i). ===================================================================== Test the input parameters Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static doublereal c_b8 = 0.; static doublereal c_b14 = -1.; /* System generated locals */ integer i__1, i__2; /* Local variables */ extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *, integer *); static doublereal taui; extern /* Subroutine */ int dspr2_(char *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *); static integer i__; static doublereal alpha; extern logical lsame_(char *, char *); extern /* Subroutine */ int daxpy_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *), dspmv_(char *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *); static integer i1; static logical upper; static integer ii; extern /* Subroutine */ int dlarfg_(integer *, doublereal *, doublereal *, integer *, doublereal *), xerbla_(char *, integer *); static integer i1i1; --tau; --e; --d__; --ap; /* Function Body */ *info = 0; upper = lsame_(uplo, "U"); if (! upper && ! lsame_(uplo, "L")) { *info = -1; } else if (*n < 0) { *info = -2; } if (*info != 0) { i__1 = -(*info); xerbla_("DSPTRD", &i__1); return 0; } /* Quick return if possible */ if (*n <= 0) { return 0; } if (upper) { /* Reduce the upper triangle of A. I1 is the index in AP of A(1,I+1). */ i1 = *n * (*n - 1) / 2 + 1; for (i__ = *n - 1; i__ >= 1; --i__) { /* Generate elementary reflector H(i) = I - tau * v * v' to annihilate A(1:i-1,i+1) */ dlarfg_(&i__, &ap[i1 + i__ - 1], &ap[i1], &c__1, &taui); e[i__] = ap[i1 + i__ - 1]; if (taui != 0.) { /* Apply H(i) from both sides to A(1:i,1:i) */ ap[i1 + i__ - 1] = 1.; /* Compute y := tau * A * v storing y in TAU(1:i) */ dspmv_(uplo, &i__, &taui, &ap[1], &ap[i1], &c__1, &c_b8, &tau[ 1], &c__1); /* Compute w := y - 1/2 * tau * (y'*v) * v */ alpha = taui * -.5 * ddot_(&i__, &tau[1], &c__1, &ap[i1], & c__1); daxpy_(&i__, &alpha, &ap[i1], &c__1, &tau[1], &c__1); /* Apply the transformation as a rank-2 update: A := A - v * w' - w * v' */ dspr2_(uplo, &i__, &c_b14, &ap[i1], &c__1, &tau[1], &c__1, & ap[1]); ap[i1 + i__ - 1] = e[i__]; } d__[i__ + 1] = ap[i1 + i__]; tau[i__] = taui; i1 -= i__; /* L10: */ } d__[1] = ap[1]; } else { /* Reduce the lower triangle of A. II is the index in AP of A(i,i) and I1I1 is the index of A(i+1,i+1). */ ii = 1; i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { i1i1 = ii + *n - i__ + 1; /* Generate elementary reflector H(i) = I - tau * v * v' to annihilate A(i+2:n,i) */ i__2 = *n - i__; dlarfg_(&i__2, &ap[ii + 1], &ap[ii + 2], &c__1, &taui); e[i__] = ap[ii + 1]; if (taui != 0.) { /* Apply H(i) from both sides to A(i+1:n,i+1:n) */ ap[ii + 1] = 1.; /* Compute y := tau * A * v storing y in TAU(i:n-1) */ i__2 = *n - i__; dspmv_(uplo, &i__2, &taui, &ap[i1i1], &ap[ii + 1], &c__1, & c_b8, &tau[i__], &c__1); /* Compute w := y - 1/2 * tau * (y'*v) * v */ i__2 = *n - i__; alpha = taui * -.5 * ddot_(&i__2, &tau[i__], &c__1, &ap[ii + 1], &c__1); i__2 = *n - i__; daxpy_(&i__2, &alpha, &ap[ii + 1], &c__1, &tau[i__], &c__1); /* Apply the transformation as a rank-2 update: A := A - v * w' - w * v' */ i__2 = *n - i__; dspr2_(uplo, &i__2, &c_b14, &ap[ii + 1], &c__1, &tau[i__], & c__1, &ap[i1i1]); ap[ii + 1] = e[i__]; } d__[i__] = ap[ii]; tau[i__] = taui; ii = i1i1; /* L20: */ } d__[*n] = ap[ii]; } return 0; /* End of DSPTRD */ } /* dsptrd_ */