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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dsytrf (f07md)

## Purpose

nag_lapack_dsytrf (f07md) computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix.

## Syntax

[a, ipiv, info] = f07md(uplo, a, 'n', n)
[a, ipiv, info] = nag_lapack_dsytrf(uplo, a, 'n', n)

## Description

nag_lapack_dsytrf (f07md) factorizes a real symmetric matrix A$A$, using the Bunch–Kaufman diagonal pivoting method. A$A$ is factorized as either A = PUDUTPT$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or A = PLDLTPT$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, where P$P$ is a permutation matrix, U$U$ (or L$L$) is a unit upper (or lower) triangular matrix and D$D$ is a symmetric block diagonal matrix with 1$1$ by 1$1$ and 2$2$ by 2$2$ diagonal blocks; U$U$ (or L$L$) has 2$2$ by 2$2$ unit diagonal blocks corresponding to the 2$2$ by 2$2$ blocks of D$D$. 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$A$ is in fact positive definite, no interchanges are performed and no 2$2$ by 2$2$ blocks occur in D$D$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies whether the upper or lower triangular part of A$A$ is stored and how A$A$ is to be factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of A$A$ is stored and A$A$ is factorized as PUDUTPT$PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of A$A$ is stored and A$A$ is factorized as PLDLTPT$PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     a(lda, : $:$) – double array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The n$n$ by n$n$ symmetric indefinite matrix A$A$.
• If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of a$a$ must be stored and the elements of the array below the diagonal are not referenced.
• If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of a$a$ must be stored and the elements of the array above the diagonal are not referenced.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the array a.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda work lwork

### Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The upper or lower triangle of A$A$ stores details of the block diagonal matrix D$D$ and the multipliers used to obtain the factor U$U$ or L$L$ as specified by uplo.
2:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the interchanges and the block structure of D$D$. More precisely,
• if ipiv(i) = k > 0${\mathbf{ipiv}}\left(i\right)=k>0$, dii${d}_{ii}$ is a 1$1$ by 1$1$ pivot block and the i$i$th row and column of A$A$ were interchanged with the k$k$th row and column;
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ and ipiv(i1) = ipiv(i) = l < 0${\mathbf{ipiv}}\left(i-1\right)={\mathbf{ipiv}}\left(i\right)=-l<0$,  ( di − 1,i − 1 di,i − 1 ) di,i − 1 dii
$\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& {d}_{ii}\end{array}\right)$ is a 2$2$ by 2$2$ pivot block and the (i1)$\left(i-1\right)$th row and column of A$A$ were interchanged with the l$l$th row and column;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$ and ipiv(i) = ipiv(i + 1) = m < 0${\mathbf{ipiv}}\left(i\right)={\mathbf{ipiv}}\left(i+1\right)=-m<0$,  ( dii di + 1,i ) di + 1,i di + 1,i + 1
$\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ is a 2$2$ by 2$2$ pivot block and the (i + 1)$\left(i+1\right)$th row and column of A$A$ were interchanged with the m$m$th row and column.
3:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: a, 4: lda, 5: ipiv, 6: work, 7: lwork, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, d(i,i)$d\left(i,i\right)$ is exactly zero. The factorization has been completed, but the block diagonal matrix D$D$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

## Accuracy

If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the computed factors U$U$ and D$D$ are the exact factors of a perturbed matrix A + E$A+E$, where
 |E| ≤ c(n)εP|U||D||UT|PT , $|E|≤c(n)εP|U||D||UT|PT ,$
c(n)$c\left(n\right)$ is a modest linear function of n$n$, and ε$\epsilon$ is the machine precision.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, a similar statement holds for the computed factors L$L$ and D$D$.

The elements of D$D$ overwrite the corresponding elements of A$A$; if D$D$ has 2$2$ by 2$2$ blocks, only the upper or lower triangle is stored, as specified by uplo.
The unit diagonal elements of U$U$ or L$L$ and the 2$2$ by 2$2$ unit diagonal blocks are not stored. The remaining elements of U$U$ or L$L$ are stored in the corresponding columns of the array a, but additional row interchanges must be applied to recover U$U$ or L$L$ explicitly (this is seldom necessary). If ipiv(i) = i${\mathbf{ipiv}}\left(\mathit{i}\right)=\mathit{i}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$ (as is the case when A$A$ is positive definite), then U$U$ or L$L$ is stored explicitly (except for its unit diagonal elements which are equal to 1$1$).
The total number of floating point operations is approximately (1/3)n3$\frac{1}{3}{n}^{3}$.
A call to nag_lapack_dsytrf (f07md) may be followed by calls to the functions:
The complex analogues of this function are nag_lapack_zhetrf (f07mr) for Hermitian matrices and nag_lapack_zsytrf (f07nr) for symmetric matrices.

## Example

```function nag_lapack_dsytrf_example
uplo = 'L';
a = [2.07, 0, 0, 0;
3.87, -0.21, 0, 0;
4.2, 1.87, 1.15, 0;
-1.15, 0.63, 2.06, -1.81];
[aOut, ipiv, info] = nag_lapack_dsytrf(uplo, a)
```
```

aOut =

2.0700         0         0         0
4.2000    1.1500         0         0
0.2230    0.8115   -2.5907         0
0.6537   -0.5960    0.3031    0.4074

ipiv =

-3
-3
3
4

info =

0

```
```function f07md_example
uplo = 'L';
a = [2.07, 0, 0, 0;
3.87, -0.21, 0, 0;
4.2, 1.87, 1.15, 0;
-1.15, 0.63, 2.06, -1.81];
[aOut, ipiv, info] = f07md(uplo, a)
```
```

aOut =

2.0700         0         0         0
4.2000    1.1500         0         0
0.2230    0.8115   -2.5907         0
0.6537   -0.5960    0.3031    0.4074

ipiv =

-3
-3
3
4

info =

0

```