﻿ f07he Method
f07he solves a real symmetric positive definite band system of linear equations with multiple right-hand sides,
 $AX=B,$
where $A$ has been factorized by f07hd.

Syntax

C#
```public static void f07he(
string uplo,
int n,
int kd,
int nrhs,
double[,] ab,
double[,] b,
out int info
)```
Visual Basic
```Public Shared Sub f07he ( _
uplo As String, _
n As Integer, _
kd As Integer, _
nrhs As Integer, _
ab As Double(,), _
b As Double(,), _
<OutAttribute> ByRef info As Integer _
)```
Visual C++
```public:
static void f07he(
String^ uplo,
int n,
int kd,
int nrhs,
array<double,2>^ ab,
array<double,2>^ b,
[OutAttribute] int% info
)```
F#
```static member f07he :
uplo : string *
n : int *
kd : int *
nrhs : int *
ab : float[,] *
b : float[,] *
info : int byref -> unit
```

Parameters

uplo
Type: System..::..String
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{"U"}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{"L"}$
$A=L{L}^{\mathrm{T}}$, 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: ${\mathbf{kd}}\ge 0$.
nrhs
Type: System..::..Int32
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
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 Cholesky factor of $A$, as returned by f07hd.
b
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: $\mathrm{dim1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: the $n$ by $r$ solution matrix $X$.
info
Type: System..::..Int32%
On exit: ${\mathbf{info}}=0$ unless the method detects an error (see [Error Indicators and Warnings]).

Description

f07he is used to solve a real symmetric positive definite band system of linear equations $AX=B$, the method must be preceded by a call to f07hd which computes the Cholesky factorization of $A$. The solution $X$ is computed by forward and backward substitution.
If ${\mathbf{uplo}}=\text{"U"}$, $A={U}^{\mathrm{T}}U$, where $U$ is upper triangular; the solution $X$ is computed by solving ${U}^{\mathrm{T}}Y=B$ and then $UX=Y$.
If ${\mathbf{uplo}}=\text{"L"}$, $A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular; the solution $X$ is computed by solving $LY=B$ and then ${L}^{\mathrm{T}}X=Y$.

References

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, LDB) 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.
${\mathbf{ifail}}=-9000$
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

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
• if ${\mathbf{uplo}}=\text{"U"}$, $\left|E\right|\le c\left(k+1\right)\epsilon \left|{U}^{\mathrm{T}}\right|\left|U\right|$;
• if ${\mathbf{uplo}}=\text{"L"}$, $\left|E\right|\le c\left(k+1\right)\epsilon \left|L\right|\left|{L}^{\mathrm{T}}\right|$,
$c\left(k+1\right)$ is a modest linear function of $k+1$, and $\epsilon$ is the machine precision.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞x∞≤ck+1condA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling (F07HHF not in this release), and an estimate for ${\kappa }_{\infty }\left(A\right)$ ($\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling (F07HGF not in this release).

Parallelism and Performance

None.

The total number of floating-point operations is approximately $4nkr$, assuming $n\gg k$.