.TH DLATRD 1 "November 2006" " LAPACK auxiliary routine (version 3.1) " " LAPACK auxiliary routine (version 3.1) " .SH NAME DLATRD - NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q\(aq * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A .SH SYNOPSIS .TP 19 SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER LDA, LDW, N, NB .TP 19 .ti +4 DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * ) .SH PURPOSE DLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q\(aq * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. If UPLO = \(aqU\(aq, DLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; .br if UPLO = \(aqL\(aq, DLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied. .br This is an auxiliary routine called by DSYTRD. .br .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. .TP 8 NB (input) INTEGER The number of rows and columns to be reduced. .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 last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = \(aqL\(aq, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal 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 >= (1,N). .TP 8 E (output) DOUBLE PRECISION array, dimension (N-1) If UPLO = \(aqU\(aq, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = \(aqL\(aq, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix. .TP 8 TAU (output) DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = \(aqU\(aq, and in TAU(1:nb) if UPLO = \(aqL\(aq. See Further Details. W (output) DOUBLE PRECISION array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A. .TP 8 LDW (input) INTEGER The leading dimension of the array W. LDW >= max(1,N). .SH FURTHER DETAILS If UPLO = \(aqU\(aq, the matrix Q is represented as a product of elementary reflectors .br Q = H(n) H(n-1) . . . H(n-nb+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:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1). .br If UPLO = \(aqL\(aq, the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(nb). .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+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i). .br The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - V*W\(aq - W*V\(aq. .br The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: .br if UPLO = \(aqU\(aq: if UPLO = \(aqL\(aq: .br ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i). .br