# ?sytrd

Reduces a real symmetric matrix to tridiagonal form.

## Syntax

FORTRAN 77:

call ssytrd(uplo, n, a, lda, d, e, tau, work, lwork, info)

call dsytrd(uplo, n, a, lda, d, e, tau, work, lwork, info)

Fortran 95:

call sytrd(a, tau [,uplo] [,info])

## Description

This routine reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: `A = Q*T*QT`. The orthogonal matrix Q is not formed explicitly but is represented as a product of n-1 elementary reflectors. Routines are provided for working with Q in this representation (see Application Notes below).

This routine calls ?latrd to reduce a real symmetric matrix to tridiagonal form by an orthogonal similarity transformation.

## Input Parameters

uplo

  CHARACTER*1. Must be 'U' or 'L'.

  If `uplo = 'U'`, a stores the upper triangular part of A.

  If `uplo = 'L'`, a stores the lower triangular part of A.

n

  INTEGER. The order of the matrix A (`n ≥ 0`).

a, work

  REAL for ssytrd

  DOUBLE PRECISION for dsytrd.

  a(lda,*) is an array containing either upper or lower triangular part of the matrix A, as specified by uplo.

  The second dimension of a must be at least max(1, n).

  work is a workspace array, its dimension `max(1, lwork)`.

lda

  INTEGER. The first dimension of a; at least max(1, n).

lwork

  INTEGER. The size of the work array (`lwork ≥ n`).

  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.

  See Application Notes for the suggested value of lwork.

## Output Parameters

a

  Overwritten by the tridiagonal matrix T and details of the orthogonal matrix Q, as specified by uplo.

d, e, tau

  REAL for ssytrd

  DOUBLE PRECISION for dsytrd.

  Arrays:

  d(*) contains the diagonal elements of the matrix T.

  The dimension of d must be at least max(1, n).

  e(*) contains the off-diagonal elements of T.

  The dimension of e must be at least max(1, n-1).

  tau(*) stores further details of the orthogonal matrix Q in the first `n-1` elements. tau(n) is used as workspace.

  The dimension of tau must be at least max(1, n).

`work(1)`

  If `info=0`, on exit `work(1)` contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

info

  INTEGER.

  If `info = 0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.

Specific details for the routine sytrd interface are the following:

a

  Holds the matrix A of size (n,n).

tau

  Holds the vector of length (n-1).

d

  Holds the vector of length (n).

e

  Holds the vector of length (n-1).

uplo

  Must be 'U' or 'L'. The default value is 'U'.

## Application Notes

For better performance, try using `lwork =n*blocksize`, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.

If it is not clear how much workspace to supply, use a generous value of lwork for the first run, or set `lwork = -1`.

In first case the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (`work(1)`) for subsequent runs.

If `lwork = -1`, then the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if lwork is less than the minimal required value and is not equal to -1, then the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

The computed matrix T is exactly similar to a matrix `A+E`, where ``||E||2 = c(n)*ε*||A||2`, `c(n)`` is a modestly increasing function of n, and ε is the machine precision.

The approximate number of floating-point operations is (4/3)n3.

After calling this routine, you can call the following:

?orgtr

  to form the computed matrix Q explicitly

?ormtr

  to multiply a real matrix by Q.

The complex counterpart of this routine is ?hetrd.

Parent topic: Symmetric Eigenvalue Problems