The routine may be called by the names f07mdf, nagf_lapacklin_dsytrf or its LAPACK name dsytrf.
3Description
f07mdf factorizes a real symmetric matrix $A$, using the Bunch–Kaufman diagonal pivoting method. $A$ is factorized as either $A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'L'}$, where $P$ is a permutation matrix, $U$ (or $L$) is a unit upper (or lower) triangular matrix and $D$ is a symmetric block diagonal matrix with $1\times 1$ and $2\times 2$ diagonal blocks; $U$ (or $L$) has $2\times 2$ unit diagonal blocks corresponding to the $2\times 2$ blocks of $D$. Row and column interchanges are performed to ensure numerical stability while preserving symmetry.
This method is suitable for symmetric matrices which are not known to be positive definite. If $A$ is in fact positive definite, no interchanges are performed and no $2\times 2$ blocks occur in $D$.
4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
1: $\mathbf{uplo}$ – Character(1)Input
On entry: specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored and $A$ is factorized as $PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored and $A$ is factorized as $PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint:
${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
3: $\mathbf{a}({\mathbf{lda}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: the $n\times n$ symmetric indefinite matrix $A$.
If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: the upper or lower triangle of $A$ is overwritten by details of the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ as specified by uplo.
4: $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07mdf is called.
Note: the dimension of the array ipiv
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On exit: details of the interchanges and the block structure of $D$. More precisely,
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.
6: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{lwork}})\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}=0$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimum performance.
7: $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f07mdf is called, unless ${\mathbf{lwork}}=\mathrm{-1}$, in which case a workspace query is assumed and the routine only calculates the optimal dimension of work (using the formula given below).
Suggested value:
for optimum performance lwork should be at least ${\mathbf{n}}\times \mathit{nb}$, where $\mathit{nb}$ is the block size.
Constraint:
${\mathbf{lwork}}\ge 1$ or ${\mathbf{lwork}}=\mathrm{-1}$.
8: $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
Element $\u27e8\mathit{\text{value}}\u27e9$ of the diagonal is exactly zero.
The factorization has been completed, but the block diagonal matrix $D$
is exactly singular, and division by zero will occur if it is
used to solve a system of equations.
7Accuracy
If ${\mathbf{uplo}}=\text{'U'}$, the computed factors $U$ and $D$ are the exact factors of a perturbed matrix $A+E$, where
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon $ is the machine precision.
If ${\mathbf{uplo}}=\text{'L'}$, a similar statement holds for the computed factors $L$ and $D$.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07mdf 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 elements of $D$ overwrite the corresponding elements of $A$; if $D$ has $2\times 2$ blocks, only the upper or lower triangle is stored, as specified by uplo.
The unit diagonal elements of $U$ or $L$ and the $2\times 2$ unit diagonal blocks are not stored. The remaining elements of $U$ or $L$ are stored in the corresponding columns of the array a, but additional row interchanges must be applied to recover $U$ or $L$ explicitly (this is seldom necessary). If ${\mathbf{ipiv}}\left(\mathit{i}\right)=\mathit{i}$, for $\mathit{i}=1,2,\dots ,n$ (as is the case when $A$ is positive definite), then $U$ or $L$ is stored explicitly (except for its unit diagonal elements which are equal to $1$).
The total number of floating-point operations is approximately $\frac{1}{3}{n}^{3}$.
A call to f07mdf may be followed by calls to the routines: