naginterfaces.library.lapacklin.dspsv

naginterfaces.library.lapacklin.dspsv(uplo, ap, b)

dspsv computes the solution to a real system of linear equations

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

where $A$ is an $n\times n$ symmetric matrix stored in packed format and $X$ and $B$ are $n\times r$ matrices.

For full information please refer to the NAG Library document for f07pa

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f07/f07paf.html

Parameters

uplostr, length 1

If $uplo=\text{'U'}$, the upper triangle of $A$ is stored.

If $uplo=\text{'L'}$, the lower triangle of $A$ is stored.

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

The $n\times n$ symmetric matrix $A$, packed by columns.

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

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

Returns

apfloat, ndarray, shape $(n\times (n+1)/2)$

The block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $A=UD{U}^T$ or $A=LD{L}^T$ as computed by dsptrf(), stored as a packed triangular matrix in the same storage format as $A$.

ipivint, ndarray, shape $\left(n\right)$

Details of the interchanges and the block structure of $D$. More precisely,

if $ipiv[i-1]=k>0$, $d_{ii}$ is a $1\times 1$ pivot block and the $i$th row and column of $A$ were interchanged with the $k$th row and column;

if $uplo=\text{'U'}$ and $ipiv[i-2]=ipiv[i-1]=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\overline{d}}_{i,i-1}\\ {\overline{d}}_{i,i-1}& d_{ii}\end{array}\right)$ is a $2\times 2$ pivot block and the $(i-1)$th row and column of $A$ were interchanged with the $l$th row and column;

if $uplo=\text{'L'}$ and $ipiv[i-1]=ipiv\left[i\right]=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& d_{i+1,i}\\ d_{i+1,i}& d_{i+1,i+1}\end{array}\right)$ is a $2\times 2$ pivot block and the $(i+1)$th row and column of $A$ were interchanged with the $m$th row and column.

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

If no exception or warning is raised, 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 $\text{nrhs}$.

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

Warns

NagAlgorithmicWarning

(errno $i>0$)

Element $⟨value⟩$ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, so the solution could not be computed.

Notes

dspsv uses the diagonal pivoting method to factor $A$ as $A=UD{U}^T$ if $uplo=\text{'U'}$ or $A=LD{L}^T$ if $uplo=\text{'L'}$, where $U$ (or $L$) is a product of permutation and unit upper (lower) triangular matrices, $D$ is symmetric and block diagonal with $1\times 1$ and $2\times 2$ diagonal blocks. The factored form of $A$ is then used to solve the system of equations $AX=B$.

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

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

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