ZHPTRD(1) LAPACK routine (version 3.2) ZHPTRD(1)NAME
ZHPTRD - reduces a complex Hermitian matrix A stored in packed form to
real symmetric tridiagonal form T by a unitary similarity transforma‐
tion
SYNOPSIS
SUBROUTINE ZHPTRD( UPLO, N, AP, D, E, TAU, INFO )
CHARACTER UPLO
INTEGER INFO, N
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 AP( * ), TAU( * )
PURPOSE
ZHPTRD reduces a complex Hermitian matrix A stored in packed form to
real symmetric tridiagonal form T by a unitary similarity transforma‐
tion: Q**H * 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) COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian 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 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 subdiag‐
onal, with the array TAU, represent the unitary 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) COMPLEX*16 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 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 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 AP, overwriting
A(i+2:n,i), and tau is stored in TAU(i).
LAPACK routine (version 3.2) November 2008 ZHPTRD(1)
