naginterfaces.library.lapacklin.dpftrs

naginterfaces.library.lapacklin.dpftrs(transr, uplo, n, ar, b)

dpftrs solves a real symmetric positive definite system of linear equations with multiple right-hand sides,

$$AX=B\text{,}$$

using the Cholesky factorization computed by dpftrf() stored in Rectangular Full Packed (RFP) format.

For full information please refer to the NAG Library document for f07we

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07wef.html

Parameters

transrstr, length 1

Specifies whether the RFP representation of $A$ is normal or transposed.

$transr=\text{'N'}$

The matrix $A$ is stored in normal RFP format.

$transr=\text{'T'}$

The matrix $A$ is stored in transposed RFP format.

uplostr, length 1

Specifies how $A$ has been factorized.

$uplo=\text{'U'}$

$A=U^{T}U$, where $U$ is upper triangular.

$uplo=\text{'L'}$

$A=L{L}^T$, where $L$ is lower triangular.

nint

$n$, the order of the matrix $A$.

arfloat, array-like, shape $(n\times (n+1)/2)$

The Cholesky factorization of $A$ stored in RFP format, as returned by dpftrf().

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

The $n\times r$ right-hand side matrix $B$.

Returns

bfloat, ndarray, shape $(n,\text{nrhs})$

The $n\times r$ solution matrix $X$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $transr$.

Constraint: $transr=\text{'N'}$ or $\text{'T'}$.

(errno $-2$)

On entry, error in parameter $uplo$.

Constraint: $uplo=\text{'U'}$ or $\text{'L'}$.

(errno $-3$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

(errno $-4$)

On entry, error in parameter $\text{nrhs}$.

Constraint: $\text{nrhs}\ge 0$.

Notes

dpftrs is used to solve a real symmetric positive definite system of linear equations $AX=B$, the function must be preceded by a call to dpftrf() which computes the Cholesky factorization of $A$, stored in RFP format. The RFP storage format is described in the F07 Introduction. The solution $X$ is computed by forward and backward substitution.

If $uplo=\text{'U'}$, $A=U^{T}U$, where $U$ is upper triangular; the solution $X$ is computed by solving $U^{T}Y=B$ and then $UX=Y$.

If $uplo=\text{'L'}$, $A=L{L}^T$, where $L$ is lower triangular; the solution $X$ is computed by solving $LY=B$ and then $L^{T}X=Y$.

References

Gustavson, F G, Waśniewski, J, Dongarra, J J and Langou, J, 2010, Rectangular full packed format for Cholesky’s algorithm: factorization, solution, and inversion, ACM Trans. Math. Software (37, 2)