.TH DSYTD2 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME DSYTD2 - a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation .SH SYNOPSIS .TP 19 SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER INFO, LDA, N .TP 19 .ti +4 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ) .SH PURPOSE DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q\(aq * A * Q = T. .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: .br = \(aqU\(aq: Upper triangular .br = \(aqL\(aq: Lower triangular .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = \(aqU\(aq, 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 = \(aqL\(aq, 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 = \(aqU\(aq, 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 = \(aqL\(aq, 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). .TP 8 D (output) DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). .TP 8 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 = \(aqU\(aq, E(i) = A(i+1,i) if UPLO = \(aqL\(aq. .TP 8 TAU (output) DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details). .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value. .SH FURTHER DETAILS If UPLO = \(aqU\(aq, the matrix Q is represented as a product of elementary reflectors .br Q = H(n-1) . . . H(2) H(1). .br Each H(i) has the form .br H(i) = I - tau * v * v\(aq .br where tau is a real scalar, and v is a real vector with .br v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in .br A(1:i-1,i+1), and tau in TAU(i). .br If UPLO = \(aqL\(aq, the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(n-1). .br Each H(i) has the form .br H(i) = I - tau * v * v\(aq .br where tau is a real scalar, and v is a real vector with .br 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). .br The contents of A on exit are illustrated by the following examples with n = 5: .br if UPLO = \(aqU\(aq: if UPLO = \(aqL\(aq: .br ( 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). .br