# NAG FL Interfacef01mef (real_​mod_​chol_​perturbed_​a)

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## 1Purpose

f01mef computes the positive definite perturbed matrix $A+E$ from the factors of a modified Cholesky factorization of a real symmetric matrix, $A$.

## 2Specification

Fortran Interface
 Subroutine f01mef ( uplo, n, a, lda, ipiv,
 Integer, Intent (In) :: n, lda, ipiv(n) Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: offdiag(n) Real (Kind=nag_wp), Intent (Inout) :: a(lda,*) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f01mef_ (const char *uplo, const Integer *n, double a[], const Integer *lda, const double offdiag[], const Integer ipiv[], Integer *ifail, const Charlen length_uplo)
The routine may be called by the names f01mef or nagf_matop_real_mod_chol_perturbed_a.

## 3Description

f01mef computes the positive definite perturbed matrix $A+E$ from the factors provided by a previous call to f01mdf. For a symmetric, possibly indefinite matrix $A$, f01mdf finds the Cheng–Higham modified Cholesky factorization
 $PT(A+E)P=LDLT ,$
when ${\mathbf{uplo}}=\text{'L'}$. Here $L$ is a unit lower triangular matrix, $P$ is a permutation matrix, $D$ is a symmetric block diagonal matrix (with blocks of order $1$ or $2$). The matrix $E$ is not explicitly formed.
If ${\mathbf{uplo}}=\text{'U'}$, we compute $A+E$ from the factorization ${P}^{\mathrm{T}}\left(A+E\right)P=UD{U}^{\mathrm{T}}$, where $U$ is a unit upper triangular matrix.
Cheng S H and Higham N J (1998) A modified Cholesky algorithm based on a symmetric indefinite factorization SIAM J. Matrix Anal. Appl. 19(4) 1097–1110

## 5Arguments

1: $\mathbf{uplo}$Character(1) Input
On entry: indicates whether the upper or lower triangular part of $A$ was stored and how it was factorized.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ was stored and we compute $A+E$ such that ${P}^{\mathrm{T}}\left(A+E\right)P=UD{U}^{\mathrm{T}}$.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ was stored and we compute $A+E$ such that ${P}^{\mathrm{T}}\left(A+E\right)P=LD{L}^{\mathrm{T}}$.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}>0$.
3: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the modified Cholesky factor of $A$, as returned by f01mdf.
On exit:
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A+E$ is returned and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A+E$ is returned and the elements of the array above the diagonal are not referenced.
4: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f01mef is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
5: $\mathbf{offdiag}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the array offdiag as returned by f01mdf.
6: $\mathbf{ipiv}\left({\mathbf{n}}\right)$Integer array Input
On entry: the array ipiv as returned by f01mdf.
7: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{uplo}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{lda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

See Section 7 in f01mdf for an indication of the accuracy of the computed factors $L$, $U$, and $D$.

## 8Parallelism and Performance

Arrays are internally allocated by f01mef. The total size of these arrays does not exceed ${n}^{2}$ real elements. All allocated memory is freed before return of f01mef.