---
name: "chetd2"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/chetd2/3lapack"
---

# chetd2(3lapack)

reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation

## NAME

CHETD2 - reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation

## SYNOPSIS

- SUBROUTINE CHETD2(

  UPLO, N, A, LDA, D, E, TAU, INFO )

- CHARACTER

  UPLO

- INTEGER

  INFO, LDA, N

- REAL

  D( \* ), E( \* )

- COMPLEX

  A( LDA, \* ), TAU( \* )

## PURPOSE

CHETD2 reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q' \* A \* Q = T.\

## ARGUMENTS

- UPLO (input) CHARACTER\*1

  Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:\
   \= 'U':  Upper triangular\
   \= 'L':  Lower triangular

- N (input) INTEGER

  The order of the matrix A.  N \>= 0.

- A (input/output) COMPLEX array, dimension (LDA,N)

  On entry, the Hermitian matrix A.  If UPLO = 'U', 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 = 'L', 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 = '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 unitary 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 unitary 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) REAL array, dimension (N)

  The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).

- E (output) REAL 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) COMPLEX 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 complex scalar, and v is a complex 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 = '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 complex scalar, and v is a complex 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 = 'U':                       if UPLO = 'L':\
\
   (  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).\