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

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

real_posdef_vband_solve computes the approximate solution of a system of real linear equations with multiple right-hand sides, $AX=B$, where $A$ 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://support.nag.com/numeric/nl/nagdoc_30/flhtml/f04/f04mcf.html

Parameters

alfloat, array-like, shape $\left(\text{lal}\right)$

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

dfloat, array-like, shape $(:)$

Note: the required length for this argument is determined as follows: if $iselct\ge 4$: $1$; otherwise: $n$.

The diagonal elements of the diagonal matrix $D$. $d$ is not referenced if $iselct\ge 4$.

nrowint, array-like, shape $\left(n\right)$

$nrow[i-1]$ must contain the width of row $i$ of $L$, i.e., the number of elements between the first (leftmost) nonzero element and the element on the diagonal, inclusive.

bfloat, array-like, shape $(n,\text{ir})$

The $n\times r$ right-hand side matrix $B$. See also Further Comments.

iselctint

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

$iselct=1$

Solve $LD{L}^{T}X=B$.

$iselct=2$

Solve $LDX=B$.

$iselct=3$

Solve $D{L}^{T}X=B$.

$iselct=4$

Solve $L{L}^{T}X=B$.

$iselct=5$

Solve $LX=B$.

$iselct=6$

Solve $L^{T}X=B$.

Returns

xfloat, ndarray, shape $(n,\text{ir})$

The $n\times r$ solution matrix $X$. See also Further Comments.

Raises

NagValueError

(errno $1$)

On entry, $\text{lal}=⟨value⟩$ and $nrow\left[0\right]+\cdots +nrow[n-1]=⟨value⟩$.

Constraint: $\text{lal}\ge nrow\left[0\right]+\cdots +nrow[n-1]$.

(errno $1$)

On entry, $I=⟨value⟩$ and $nrow[I-1]=⟨value⟩$.

Constraint: $nrow[I-1]\ge 1$ and $nrow[I-1]\le I$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

On entry, $\text{ir}=⟨value⟩$.

Constraint: $\text{ir}\ge 1$.

(errno $3$)

On entry, $iselct=⟨value⟩$.

Constraint: $iselct\ge 1$ and $iselct\le 6$.

(errno $4$)

On entry, $I=⟨value⟩$.

Constraint: $d[I-1]\ne 0.0$.

(errno $5$)

At least one diagonal entry of $al$ 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 $AX=B$, following a call of matop.real_vband_posdef_fac to determine the Cholesky factorization $A=LD{L}^T$ of the symmetric positive definite variable-bandwidth matrix $A$.

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

1. $LD{L}^{T}X=B$ (usual system),

2. $LDX=B$ (lower triangular system),

3. $D{L}^{T}X=B$ (upper triangular system),

4. $L{L}^{T}X=B$

5. $LX=B$ (unit lower triangular system),

6. $L^{T}X=B$ (unit upper triangular system).

$L$ denotes a unit lower triangular variable-bandwidth matrix of order $n$, $D$ a diagonal matrix of order $n$, and $B$ a set of right-hand sides.

The matrix $L$ is represented by the elements lying within its envelope, i.e., between the first nonzero of each row and the diagonal. The width $nrow[i-1]$ of the $i$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