naginterfaces.library.linsys.complex_​posdef_​tridiag_​solve¶

naginterfaces.library.linsys.complex_posdef_tridiag_solve(d, e, b)[source]

complex_posdef_tridiag_solve computes the solution to a complex system of linear equations , where is an Hermitian positive definite tridiagonal matrix and and are matrices. An estimate of the condition number of and an error bound for the computed solution are also returned.

For full information please refer to the NAG Library document for f04cg

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/f04/f04cgf.html

Parameters
dfloat, array-like, shape

Must contain the diagonal elements of the tridiagonal matrix .

ecomplex, array-like, shape

Must contain the subdiagonal elements of the tridiagonal matrix .

bcomplex, array-like, shape

The matrix of right-hand sides .

Returns
dfloat, ndarray, shape

If the function exits successfully or = + 1, is overwritten by the diagonal elements of the diagonal matrix from the factorization of .

ecomplex, ndarray, shape

If the function exits successfully or = + 1, is overwritten by the subdiagonal elements of the unit lower bidiagonal matrix from the factorization of . ( can also be regarded as the conjugate of the superdiagonal of the unit upper bidiagonal factor from the factorization of .)

bcomplex, ndarray, shape

If the function exits successfully or = + 1, the solution matrix .

rcondfloat

If the function exits successfully or = + 1, an estimate of the reciprocal of the condition number of the matrix , computed as .

errbndfloat

If the function exits successfully or = + 1, an estimate of the forward error bound for a computed solution , such that , where is a column of the computed solution returned in the array and is the corresponding column of the exact solution . If is less than machine precision, is returned as unity.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

The principal minor of order of the matrix is not positive definite. The factorization has not been completed and the solution could not be computed.

Warns
NagAlgorithmicWarning
(errno )

A solution has been computed, but is less than machine precision so that the matrix is numerically singular.

Notes

is factorized as , where is a unit lower bidiagonal matrix and is a real diagonal matrix, and the factored form of is then used to solve the system of equations.

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug

Higham, N J, 2002, Accuracy and Stability of Numerical Algorithms, (2nd Edition), SIAM, Philadelphia