?pteqr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?pteqr

Computes all eigenvalues and (optionally) all eigenvectors of a real symmetric positive-definite tridiagonal matrix.

Syntax

call spteqr(compz, n, d, e, z, ldz, work, info)

call dpteqr(compz, n, d, e, z, ldz, work, info)

call cpteqr(compz, n, d, e, z, ldz, work, info)

call zpteqr(compz, n, d, e, z, ldz, work, info)

call rpteqr(d, e [,z] [,compz] [,info])

call pteqr(d, e [,z] [,compz] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine computes all the eigenvalues and (optionally) all the eigenvectors of a real symmetric positive-definite tridiagonal matrix T. In other words, the routine can compute the spectral factorization: T = Z*Λ*Z^T.

Here Λ is a diagonal matrix whose diagonal elements are the eigenvalues λ_i; Z is an orthogonal matrix whose columns are eigenvectors. Thus,

T*z_i = λ_i*z_i for i = 1, 2, ..., n.

(The routine normalizes the eigenvectors so that ||z_i||₂ = 1.)

You can also use the routine for computing the eigenvalues and eigenvectors of real symmetric (or complex Hermitian) positive-definite matrices A reduced to tridiagonal form T: A = Q*T*Q^H. In this case, the spectral factorization is as follows: A = Q*T*Q^H = (QZ)*Λ*(QZ)^H. Before calling ?pteqr, you must reduce A to tridiagonal form and generate the explicit matrix Q by calling the following routines:

	for real matrices:	for complex matrices:
full storage	?sytrd, ?orgtr	?hetrd, ?ungtr
packed storage	?sptrd, ?opgtr	?hptrd, ?upgtr
band storage	?sbtrd`(vect`='V')	?hbtrd`(vect`='V')

The routine first factorizes T as L*D*L^H where L is a unit lower bidiagonal matrix, and D is a diagonal matrix. Then it forms the bidiagonal matrix B = L*D^1/2 and calls ?bdsqr to compute the singular values of B, which are the square roots of the eigenvalues of T.

Input Parameters

compz

CHARACTER*1. Must be 'N' or 'I' or 'V'.

If compz = 'N', the routine computes eigenvalues only.

If compz = 'I', the routine computes the eigenvalues and eigenvectors of the tridiagonal matrix T.

If compz = 'V', the routine computes the eigenvalues and eigenvectors of A (and the array z must contain the matrix Q on entry).

n

INTEGER. The order of the matrix T (n≥ 0).

d, e, work

REAL for single-precision flavors

DOUBLE PRECISION for double-precision flavors.

Arrays:

d(*) contains the diagonal elements of T.

The size of d must be at least max(1, n).

e(*) contains the off-diagonal elements of T.

The size of e must be at least max(1, n-1).

work(*) is a workspace array.

The dimension of work must be:

at least 1 if compz = 'N';

at least max(1, 4*n-4) if compz = 'V' or 'I'.

z

REAL for spteqr

DOUBLE PRECISION for dpteqr

COMPLEX for cpteqr

DOUBLE COMPLEX for zpteqr.

Array, size (ldz,*)

If compz = 'N' or 'I', z need not be set.

If compz = 'V', z must contain the orthogonal matrix used in the reduction to tridiagonal form..

The second dimension of z must be:

at least 1 if compz = 'N';

at least max(1, n) if compz = 'V' or 'I'.

ldz

INTEGER. The leading dimension of z. Constraints:

ldz≥ 1 if compz = 'N';

ldz≥ max(1, n) if compz = 'V' or 'I'.

Output Parameters

d

The n eigenvalues in descending order, unless info > 0.

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?pteqr