F02GBF computes all the eigenvalues, and optionally all the eigenvectors, of a complex general matrix.
F02GBF computes all the eigenvalues, and optionally all the right eigenvectors, of a complex general matrix
A:
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,
ε is the
machine precision, and
si is the reciprocal condition number of
λi;
A′ is the balanced form of the original matrix
A (see
Section 8), and
A′≤A.
If
xi is the corresponding exact eigenvector, and
x~i is the corresponding computed eigenvector, then the angle
θx~i,xi between them is bounded as follows:
where
sepi is the reciprocal condition number of
xi.
The condition numbers
si and
sepi may be computed by calling
F08QYF (ZTRSNA), using the Schur form of the balanced matrix
A′ which is returned in the array
A when
JOB='V'.
F02GBF calls routines from LAPACK in
Chapter F08. It first balances the matrix, using a diagonal similarity transformation to reduce its norm; and then reduces the balanced matrix
A′ to upper Hessenberg form
H, using a unitary similarity transformation:
A′=QHQH. If only eigenvalues are required, the routine uses the Hessenberg
QR algorithm to compute the eigenvalues. If the eigenvectors are required, the routine first forms the unitary matrix
Q that was used in the reduction to Hessenberg form; it then uses the Hessenberg
QR algorithm to compute the Schur factorization of
A′ as
A′=ZTZH. It computes the right eigenvectors of
T by backward substitution, pre-multiplies them by
Z to form the eigenvectors of
A′; and finally transforms the eigenvectors to those of the original matrix
A.
This example computes all the eigenvalues and eigenvectors of the matrix
A, where