naginterfaces.library.linsys.real_​posdef_​vband_​solve

naginterfaces.library.linsys.real_posdef_vband_solve(al, d, nrow, b, iselct)[source]

real_posdef_vband_solve computes the approximate solution of a system of real linear equations with multiple right-hand sides, , where is a symmetric positive definite variable-bandwidth matrix, which has previously been factorized by matop.real_vband_posdef_fac. Related systems may also be solved.

For full information please refer to the NAG Library document for f04mc

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/f04/f04mcf.html

Parameters
alfloat, array-like, shape

The elements within the envelope of the lower triangular matrix , taken in row by row order, as returned by matop.real_vband_posdef_fac. The unit diagonal elements of must be stored explicitly.

dfloat, array-like, shape

Note: the required length for this argument is determined as follows: if : ; otherwise: .

The diagonal elements of the diagonal matrix . is not referenced if .

nrowint, array-like, shape

must contain the width of row of , i.e., the number of elements between the first (leftmost) nonzero element and the element on the diagonal, inclusive.

bfloat, array-like, shape

The right-hand side matrix . See also Further Comments.

iselctint

Must specify the type of system to be solved, as follows:

Solve .

Solve .

Solve .

Solve .

Solve .

Solve .

Returns
xfloat, ndarray, shape

The solution matrix . See also Further Comments.

Raises
NagValueError
(errno )

On entry, and .

Constraint: .

(errno )

On entry, and .

Constraint: and .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: and .

(errno )

On entry, .

Constraint: .

(errno )

At least one diagonal entry of is not unit.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

The normal use of this function is the solution of the systems , following a call of matop.real_vband_posdef_fac to determine the Cholesky factorization of the symmetric positive definite variable-bandwidth matrix .

However, the function may be used to solve any one of the following systems of linear algebraic equations:

  1. (usual system),

  2. (lower triangular system),

  3. (upper triangular system),

  4. (unit lower triangular system),

  5. (unit upper triangular system).

denotes a unit lower triangular variable-bandwidth matrix of order , a diagonal matrix of order , and a set of right-hand sides.

The matrix is represented by the elements lying within its envelope, i.e., between the first nonzero of each row and the diagonal. The width of the th row is the number of elements between the first nonzero element and the element on the diagonal inclusive.

References

Wilkinson, J H and Reinsch, C, 1971, Handbook for Automatic Computation II, Linear Algebra, Springer–Verlag