F02FAF computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric matrix.
F02FAF computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric matrix
A:
In other words, it computes the spectral factorization of
A:
where
Λ is a diagonal matrix whose diagonal elements are the eigenvalues
λi, and
Z is an orthogonal matrix, whose columns are the eigenvectors
zi.
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
If
λi is an exact eigenvalue, and
λ~i is the corresponding computed value, then
where
cn is a modestly increasing function of
n, and
ε is the
machine precision.
If
zi is the corresponding exact eigenvector, and
z~i is the corresponding computed eigenvector, then the angle
θz~i,zi between them is bounded as follows:
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.
F02FAF calls routines from LAPACK in
Chapter F08. It first reduces
A to tridiagonal form
T, using an orthogonal similarity transformation:
A=QTQT. If only eigenvalues are required, the routine uses a root-free variant of the symmetric tridiagonal
QR algorithm. If eigenvectors are required, the routine first forms the orthogonal matrix
Q that was used in the reduction to tridiagonal form; it then uses the symmetric tridiagonal
QR algorithm to reduce
T to
Λ, using a further orthogonal transformation:
T=SΛST; and at the same time accumulates the matrix
Z=QS.
This example computes all the eigenvalues and eigenvectors of the matrix
A, where