﻿ f07hd Method
f07hd computes the Cholesky factorization of a real symmetric positive definite band matrix.

# Syntax

C#
```public static void f07hd(
string uplo,
int n,
int kd,
double[,] ab,
out int info
)```
Visual Basic
```Public Shared Sub f07hd ( _
uplo As String, _
n As Integer, _
kd As Integer, _
ab As Double(,), _
<OutAttribute> ByRef info As Integer _
)```
Visual C++
```public:
static void f07hd(
String^ uplo,
int n,
int kd,
array<double,2>^ ab,
[OutAttribute] int% info
)```
F#
```static member f07hd :
uplo : string *
n : int *
kd : int *
ab : float[,] *
info : int byref -> unit
```

#### Parameters

uplo
Type: System..::..String
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 ${U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{"L"}$
The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{"U"}$ or $\text{"L"}$.
n
Type: System..::..Int32
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
kd
Type: System..::..Int32
On entry: ${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix $A$.
Constraint: ${\mathbf{kd}}\ge 0$.
ab
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: $\mathrm{dim1}\ge {\mathbf{kd}}+1$
Note: the second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ symmetric positive definite band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{uplo}}=\text{"U"}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left[{k}_{d}+1+i-j,j\right]\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{"L"}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left[1+i-j,j\right]\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
On exit: the upper or lower triangle of $A$ is overwritten by the Cholesky factor $U$ or $L$ as specified by uplo, using the same storage format as described above.
info
Type: System..::..Int32%
On exit: ${\mathbf{info}}=0$ unless the method detects an error (see [Error Indicators and Warnings]).

# Description

f07hd forms the Cholesky factorization of a real symmetric positive definite band matrix $A$ either as $A={U}^{\mathrm{T}}U$ if ${\mathbf{uplo}}=\text{"U"}$ or $A=L{L}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{"L"}$, where $U$ (or $L$) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as $A$.

# References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

# Error Indicators and Warnings

Some error messages may refer to parameters that are dropped from this interface (LDAB) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
The leading minor of order $〈\mathit{\text{value}}〉$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$. There is no method specifically designed to factorize a symmetric band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling f07bd or as a full symmetric matrix, by calling f07md.
${\mathbf{ifail}}=-9000$
An error occured, see message report.
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-4000$
Invalid dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-8000$
Negative dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$

# Accuracy

If ${\mathbf{uplo}}=\text{"U"}$, the computed factor $U$ is the exact factor of a perturbed matrix $A+E$, where
 $E≤ck+1εUTU,$
$c\left(k+1\right)$ is a modest linear function of $k+1$, and $\epsilon$ is the machine precision.
If ${\mathbf{uplo}}=\text{"L"}$, a similar statement holds for the computed factor $L$. It follows that $\left|{e}_{ij}\right|\le c\left(k+1\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.

# Parallelism and Performance

None.

The total number of floating-point operations is approximately $n{\left(k+1\right)}^{2}$, assuming $n\gg k$.
A call to f07hd may be followed by calls to the methods:
• f07he to solve $AX=B$;
• (F07HGF not in this release) to estimate the condition number of $A$.
The complex analogue of this method is (F07HRF not in this release).

# Example

This example computes the Cholesky factorization of the matrix $A$, where
 $A=5.492.680.000.002.685.63-2.390.000.00-2.392.60-2.220.000.00-2.225.17.$

Example program (C#): f07hde.cs

Example program data: f07hde.d

Example program results: f07hde.r