---
name: "dsptrd"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/dsptrd/3lapack"
---

# dsptrd(3lapack)

reduces a real symmetric matrix A stored in packed form to symmetric tridiagonal form T by an orthogonal similarity transformation

## NAME

DSPTRD - reduces a real symmetric matrix A stored in packed form to symmetric tridiagonal form T by an orthogonal similarity transformation

## SYNOPSIS

- SUBROUTINE DSPTRD(

  UPLO, N, AP, D, E, TAU, INFO )

- CHARACTER

  UPLO

- INTEGER

  INFO, N

- DOUBLE

  PRECISION AP( \* ), D( \* ), E( \* ), TAU( \* )

## 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).\