# dsytrd (l) - Linux Man Pages

## NAME

DSYTRD - reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation

## SYNOPSIS

SUBROUTINE DSYTRD(
UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )

CHARACTER UPLO

INTEGER INFO, LDA, LWORK, N

DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ), WORK( * )

## PURPOSE

DSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.

## ARGUMENTS

UPLO (input) CHARACTER*1
= aqUaq: Upper triangle of A is stored;
= aqLaq: Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = aqUaq, 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 = aqLaq, 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 = aqUaq, 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 = aqLaq, 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) 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 = aqUaq, E(i) = A(i+1,i) if UPLO = aqLaq.
TAU (output) DOUBLE PRECISION array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further Details).
WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1. For optimum performance LWORK >= N*NB, where NB is the optimal blocksize. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

## FURTHER DETAILS

If UPLO = aqUaq, the matrix Q is represented as a product of elementary reflectors

H(n-1) . . . H(2) H(1).
Each H(i) has the form

H(i) I - tau vaq
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 = aqLaq, the matrix Q is represented as a product of elementary reflectors

H(1) H(2) . . . H(n-1).
Each H(i) has the form

H(i) I - tau vaq
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 = aqUaq: if UPLO = aqLaq:

v2  v3  v4                                )
v3  v4                              )
v4               v1             )
v1  v2         )
v1  v2  v3     ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).