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

# zlatrd(3lapack)

reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q' \* A \* Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A

## NAME

ZLATRD - reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q' \* A \* Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A

## SYNOPSIS

- SUBROUTINE ZLATRD(

  UPLO, N, NB, A, LDA, E, TAU, W, LDW )

- CHARACTER

  UPLO

- INTEGER

  LDA, LDW, N, NB

- DOUBLE

  PRECISION E( \* )

- COMPLEX\*16

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

## PURPOSE

ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q' \* A \* Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied;\
 if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.\
 This is an auxiliary routine called by ZHETRD.\

## 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.

- NB (input) INTEGER

  The number of rows and columns to be reduced.

- A (input/output) COMPLEX\*16 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 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 unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', 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  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).

- E (output) DOUBLE PRECISION array, dimension (N-1)

  If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.

- TAU (output) COMPLEX\*16 array, dimension (N-1)

  The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. See Further Details. W       (output) COMPLEX\*16 array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.

- LDW (input) INTEGER

  The leading dimension of the array W. LDW \>= max(1,N).

## FURTHER DETAILS

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors\
\
    Q = H(n) H(n-1) . . . H(n-nb+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: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).\
 If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors\
\
    Q = H(1) H(2) . . . H(nb).\
 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+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).\
 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 Hermitian rank-2k update of the form: A := A - V\*W' - W\*V'.\
 The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:\
 if UPLO = 'U':                       if UPLO = 'L':\
\
   (  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).\