$$A=UD{U}^{\mathrm{T}}\text{\hspace{1em} or \hspace{1em}}A=LD{L}^{\mathrm{T}}$$
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.
The routine may be called by the names f07pbf, nagf_lapacklin_dspsvx or its LAPACK name dspsvx.
3Description
f07pbf performs the following steps:
1.If ${\mathbf{fact}}=\text{'N'}$, the diagonal pivoting method is used to factor $A$ as $A=UD{U}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=LD{L}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'L'}$, 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 routine returns with ${\mathbf{info}}=i$. 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, ${\mathbf{info}}={\mathbf{n}+{\mathbf{1}}}$ is returned as a warning, but the routine 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.
4References
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
5Arguments
1: $\mathbf{fact}$ – Character(1)Input
On entry: specifies whether or not the factorized form of the matrix $A$ has been supplied.
${\mathbf{fact}}=\text{'F'}$
afp and ipiv contain the factorized form of the matrix $A$. afp and ipiv will not be modified.
${\mathbf{fact}}=\text{'N'}$
The matrix $A$ will be copied to afp and factorized.
Constraint:
${\mathbf{fact}}=\text{'F'}$ or $\text{'N'}$.
2: $\mathbf{uplo}$ – Character(1)Input
On entry: if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ is stored.
Constraint:
${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
4: $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{nrhs}}\ge 0$.
5: $\mathbf{ap}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array ap
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}}\times ({\mathbf{n}}+1)/2)$.
On entry: the $n\times n$ symmetric matrix $A$, packed by columns.
More precisely,
if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j(j-1)/2\right)$ for $i\le j$;
if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+(2n-j)(j-1)/2\right)$ for $i\ge j$.
6: $\mathbf{afp}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array afp
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}}\times ({\mathbf{n}}+1)/2)$.
On entry: if ${\mathbf{fact}}=\text{'F'}$, afp contains the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $A=UD{U}^{\mathrm{T}}$ or $A=LD{L}^{\mathrm{T}}$ as computed by f07pdf, stored as a packed triangular matrix in the same storage format as $A$.
On exit: if ${\mathbf{fact}}=\text{'N'}$, afp contains the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $A=UD{U}^{\mathrm{T}}$ or $A=LD{L}^{\mathrm{T}}$ as computed by f07pdf, stored as a packed triangular matrix in the same storage format as $A$.
On entry: if ${\mathbf{fact}}=\text{'F'}$, ipiv contains details of the interchanges and the block structure of $D$, as determined by f07pdf.
if ${\mathbf{ipiv}}\left(i\right)=k>0$, ${d}_{ii}$ 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 ${\mathbf{uplo}}=\text{'U'}$ and ${\mathbf{ipiv}}\left(i-1\right)={\mathbf{ipiv}}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\overline{d}}_{i,i-1}\\ {\overline{d}}_{i,i-1}& {d}_{ii}\end{array}\right)$ 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 ${\mathbf{uplo}}=\text{'L'}$ and ${\mathbf{ipiv}}\left(i\right)={\mathbf{ipiv}}\left(i+1\right)=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ 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.
On exit: if ${\mathbf{fact}}=\text{'N'}$, ipiv contains details of the interchanges and the block structure of $D$, as determined by f07pdf, as described above.
8: $\mathbf{b}({\mathbf{ldb}},*)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array b
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{nrhs}})$.
On entry: the $n\times r$ right-hand side matrix $B$.
9: $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07pbf is called.
On exit: the estimate of the reciprocal condition number of the matrix $A$. If ${\mathbf{rcond}}=0.0$, the matrix may be exactly singular. This condition is indicated by ${\mathbf{info}}>{\mathbf{0}}\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{info}}\le \mathbf{n}$. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by ${\mathbf{info}}={\mathbf{n}+{\mathbf{1}}}$.
13: $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, an estimate of the forward error bound for each computed solution vector, such that ${\Vert {\hat{x}}_{j}-{x}_{j}\Vert}_{\infty}/{\Vert {x}_{j}\Vert}_{\infty}\le {\mathbf{ferr}}\left(j\right)$ 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.
14: $\mathbf{berr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, 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).
15: $\mathbf{work}\left(3\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
Element $\u27e8\mathit{\text{value}}\u27e9$ 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.
${\mathbf{rcond}}=0.0$ is returned.
${\mathbf{info}}={\mathbf{n}}+1$
$D$ 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.
7Accuracy
For each right-hand side vector $b$, the computed solution $\hat{x}$ is the exact solution of a perturbed system of equations $(A+E)\hat{x}=b$, where
where
$\mathrm{cond}(A,\hat{x},b)={\Vert \left|{A}^{-1}\right|(\left|A\right|\left|\hat{x}\right|+\left|b\right|)\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}\le \mathrm{cond}\left(A\right)={\Vert \left|{A}^{-1}\right|\left|A\right|\Vert}_{\infty}\le {\kappa}_{\infty}\left(A\right)$.
If $\hat{x}$ is the $j$th column of $X$, then ${w}_{c}$ is returned in ${\mathbf{berr}}\left(j\right)$ and a bound on ${\Vert x-\hat{x}\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}$ is returned in ${\mathbf{ferr}}\left(j\right)$. See Section 4.4 of Anderson et al. (1999) for further details.
8Parallelism and Performance
f07pbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07pbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The factorization of $A$ requires approximately $\frac{1}{3}{n}^{3}$ floating-point operations.
For each right-hand side, computation of the backward error involves a minimum of $4{n}^{2}$ floating-point operations. Each step of iterative refinement involves an additional $6{n}^{2}$ operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2{n}^{2}$ operations.
The complex analogues of this routine are f07ppf for Hermitian matrices, and f07qpf for symmetric matrices.