clatrd (l)  Linux Man Pages
clatrd: reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Qaq * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A
NAME
CLATRD  reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Qaq * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of ASYNOPSIS
 SUBROUTINE CLATRD(
 UPLO, N, NB, A, LDA, E, TAU, W, LDW )
 CHARACTER UPLO
 INTEGER LDA, LDW, N, NB
 REAL E( * )
 COMPLEX A( LDA, * ), TAU( * ), W( LDW, * )
PURPOSE
CLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Qaq * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. If UPLO = aqUaq, CLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied;if UPLO = aqLaq, CLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.
This is an auxiliary routine called by CHETRD.
ARGUMENTS
 UPLO (input) CHARACTER*1

Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= aqUaq: Upper triangular
= aqLaq: 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 array, dimension (LDA,N)
 On entry, the Hermitian matrix A. If UPLO = aqUaq, the leading nbyn 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 nbyn 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 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 = aqLaq, 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) REAL array, dimension (N1)
 If UPLO = aqUaq, E(nnb:n1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = aqLaq, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
 TAU (output) COMPLEX array, dimension (N1)
 The scalar factors of the elementary reflectors, stored in TAU(nnb:n1) if UPLO = aqUaq, and in TAU(1:nb) if UPLO = aqLaq. See Further Details. W (output) COMPLEX array, dimension (LDW,NB) The nbynb 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 = aqUaq, the matrix Q is represented as a product of elementary reflectorsQ
Each H(i) has the form
H(i)
where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i1) = 1; v(1:i1) is stored on exit in A(1:i1,i), and tau in TAU(i1).
If UPLO = aqLaq, the matrix Q is represented as a product of elementary reflectors
Q
Each H(i) has the form
H(i)
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 nbynb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank2k update of the form: A := A  V*Waq  W*Vaq.
The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:
if UPLO = aqUaq: if UPLO = aqLaq: