naginterfaces.library.lapacklin.zpbtrs

naginterfaces.library.lapacklin.zpbtrs(uplo, kd, ab, b)

zpbtrs solves a complex Hermitian positive definite band system of linear equations with multiple right-hand sides,

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

where $A$ has been factorized by zpbtrf().

For full information please refer to the NAG Library document for f07hs

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07hsf.html

Parameters

uplostr, length 1

Specifies how $A$ has been factorized.

$uplo=\text{'U'}$

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

$uplo=\text{'L'}$

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

kdint

$k_d$, the number of superdiagonals or subdiagonals of the matrix $A$.

abcomplex, array-like, shape $(kd+1,n)$

The Cholesky factor of $A$, as returned by zpbtrf().

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

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

Returns

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

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

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $uplo$.

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

(errno $-2$)

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

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $kd$.

Constraint: $kd\ge 0$.

(errno $-4$)

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

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

Notes

zpbtrs is used to solve a complex Hermitian positive definite band system of linear equations $AX=B$, the function must be preceded by a call to zpbtrf() which computes the Cholesky factorization of $A$. The solution $X$ is computed by forward and backward substitution.

If $uplo=\text{'U'}$, $A=U^{H}U$, where $U$ is upper triangular; the solution $X$ is computed by solving $U^{H}Y=B$ and then $UX=Y$.

If $uplo=\text{'L'}$, $A=L{L}^H$, where $L$ is lower triangular; the solution $X$ is computed by solving $LY=B$ and then $L^{H}X=Y$.

References

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore