CHPTRD(3F) CHPTRD(3F)
CHPTRD - reduce a complex Hermitian matrix A stored in packed form to
real symmetric tridiagonal form T by a unitary similarity transformation
SUBROUTINE CHPTRD( UPLO, N, AP, D, E, TAU, INFO )
CHARACTER UPLO
INTEGER INFO, N
REAL D( * ), E( * )
COMPLEX AP( * ), TAU( * )
CHPTRD reduces a complex Hermitian matrix A stored in packed form to real
symmetric tridiagonal form T by a unitary similarity transformation: Q**H
* A * Q = T.
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 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 + (j1)*j/2)
= A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*nj)/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 subdiagonal, with the array TAU,
represent the unitary matrix Q as a product of elementary
reflectors. See Further Details. 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'.
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CHPTRD(3F) CHPTRD(3F)
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 AP, overwriting A(1:i1,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).
CHPTRD(3F) CHPTRD(3F)
CHPTRD - reduce a complex Hermitian matrix A stored in packed form to
real symmetric tridiagonal form T by a unitary similarity transformation
SUBROUTINE CHPTRD( UPLO, N, AP, D, E, TAU, INFO )
CHARACTER UPLO
INTEGER INFO, N
REAL D( * ), E( * )
COMPLEX AP( * ), TAU( * )
CHPTRD reduces a complex Hermitian matrix A stored in packed form to real
symmetric tridiagonal form T by a unitary similarity transformation: Q**H
* A * Q = T.
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 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 + (j1)*j/2)
= A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*nj)/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 subdiagonal, with the array TAU,
represent the unitary matrix Q as a product of elementary
reflectors. See Further Details. 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'.
Page 1
CHPTRD(3F) CHPTRD(3F)
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 AP, overwriting A(1:i1,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).
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