naginterfaces.library.lapacklin.dspsvx

naginterfaces.library.lapacklin.dspsvx(fact, uplo, nrhs, ap, afp, ipiv, b)

dspsvx uses the diagonal pivoting factorization

$$A=UD{U}^T\text{or}A=LD{L}^T$$

to compute 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. Error bounds on the solution and a condition estimate are also provided.

For full information please refer to the NAG Library document for f07pb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07pbf.html

Parameters

factstr, length 1

Specifies whether or not the factorized form of the matrix $A$ has been supplied.

$fact=\text{'F'}$

$afp$ and $ipiv$ contain the factorized form of the matrix $A$. $afp$ and $ipiv$ will not be modified.

$fact=\text{'N'}$

The matrix $A$ will be copied to $afp$ and factorized.

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.

nrhsint

$r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.

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

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

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

If $fact=\text{'F'}$, $afp$ contains 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, array-like, shape $\left(n\right)$

If $fact=\text{'F'}$, $ipiv$ contains details of the interchanges and the block structure of $D$, as determined by dsptrf().

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, array-like, shape $(n,nrhs)$

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

Returns

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

If $fact=\text{'N'}$, $afp$ contains 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 $\left(n\right)$

If $fact=\text{'N'}$, $ipiv$ contains details of the interchanges and the block structure of $D$, as determined by dsptrf(), as described above.

xfloat, ndarray, shape $(n,nrhs)$

If the function exits successfully or $errno$ = $n$ + 1, the $n\times r$ solution matrix $X$.

rcondfloat

The estimate of the reciprocal condition number of the matrix $A$. If $rcond=0.0$, the matrix may be exactly singular. This condition is indicated by $errno$ in 1 … $n$. Otherwise, if $rcond$ is less than the machine precision, the matrix is singular to working precision. This condition is indicated by $errno$ = $n$ + 1.

ferrfloat, ndarray, shape $\left(nrhs\right)$

If the function exits successfully or $errno$ = $n$ + 1, an estimate of the forward error bound for each computed solution vector, such that ${\Vert {\hat{x}}_j-x_j\Vert }_{\infty }/{\Vert x_j\Vert }_{\infty }\le ferr[j-1]$ where ${\hat{x}}_j$ is the $j$th column of the computed solution returned in the array $x$ and $x_j$ is the corresponding column of the exact solution $X$. The estimate is as reliable as the estimate for $rcond$, and is almost always a slight overestimate of the true error.

berrfloat, ndarray, shape $\left(nrhs\right)$

If the function exits successfully or $errno$ = $n$ + 1, an estimate of the component-wise relative backward error of each computed solution vector ${\hat{x}}_j$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_j$ an exact solution).

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $fact$.

Constraint: $fact=\text{'F'}$ or $\text{'N'}$.

(errno $-2$)

On entry, error in parameter $uplo$.

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

(errno $-3$)

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

Constraint: $n\ge 0$.

(errno $-4$)

On entry, error in parameter $nrhs$.

Constraint: $nrhs\ge 0$.

Warns

NagAlgorithmicWarning

(errno $(i>0)\text{and}(i\le n)$)

Element $⟨value⟩$ of the diagonal is exactly zero. The factorization has been completed, but the factor $D$ is exactly singular, so the solution and error bounds could not be computed. $rcond=0.0$ is returned.

(errno $n+1$)

$D$ is nonsingular, but $rcond$ is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of $rcond$ would suggest.

Notes

dspsvx performs the following steps:

1. If $fact=\text{'N'}$, the diagonal pivoting method is used 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 and $D$ is symmetric and block diagonal with $1\times 1$ and $2\times 2$ diagonal blocks.

2. If some $d_{ii}=0$, so that $D$ is exactly singular, then the function returns with $\text{errno}=i$. Otherwise, the factored form of $A$ is used to estimate the condition number of the matrix $A$. If the reciprocal of the condition number is less than machine precision, $errno$ = $n$ + 1 is returned as a warning, but the function still goes on to solve for $X$ and compute error bounds as described below.

3. The system of equations is solved for $X$ using the factored form of $A$.

4. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

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