# NAG CL Interfacef04bfc (real_​posdef_​band_​solve)

## 1Purpose

f04bfc computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ symmetric positive definite band matrix of band width $2k+1$, and $X$ and $B$ are $n$ by $r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

## 2Specification

 #include
 void f04bfc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, Integer nrhs, double ab[], Integer pdab, double b[], Integer pdb, double *rcond, double *errbnd, NagError *fail)
The function may be called by the names: f04bfc, nag_linsys_real_posdef_band_solve or nag_real_sym_posdef_band_lin_solve.

## 3Description

The Cholesky factorization is used to factor $A$ as $A={U}^{\mathrm{T}}U$, if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, or $A=L{L}^{\mathrm{T}}$, if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, where $U$ is an upper triangular band matrix with $k$ superdiagonals, and $L$ is a lower triangular band matrix with $k$ subdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.

## 4References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{uplo}$Nag_UploType Input
On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangle of the matrix $A$ is stored.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangle of the matrix $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3: $\mathbf{n}$Integer Input
On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{kd}$Integer Input
On entry: the number of superdiagonals $k$ (and the number of subdiagonals) of the band matrix $A$.
Constraint: ${\mathbf{kd}}\ge 0$.
5: $\mathbf{nrhs}$Integer Input
On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
6: $\mathbf{ab}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdab}}×{\mathbf{n}}\right)$.
On entry:
• if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ then
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${a}_{ij}$ is stored in ${\mathbf{ab}}\left[\left(j-1\right)×{\mathbf{pdab}}+{\mathbf{kd}}+i-j\right]$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${a}_{ij}$ is stored in ${\mathbf{ab}}\left[\left(i-1\right)×{\mathbf{pdab}}+j-i\right]$,
for $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{\mathbf{kd}}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$ then
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${a}_{ij}$ is stored in ${\mathbf{ab}}\left[\left(j-1\right)×{\mathbf{pdab}}+i-j\right]$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${a}_{ij}$ is stored in ${\mathbf{ab}}\left[\left(i-1\right)×{\mathbf{pdab}}+{\mathbf{kd}}+j-i\right]$,
for $j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{\mathbf{kd}}\right)$,
where ${\mathbf{pdab}}\ge {\mathbf{kd}}+1$ is the stride separating diagonal matrix elements in the array ab.
See Section 9 below for further details.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, the factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$, in the same storage format as $A$.
7: $\mathbf{pdab}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array ab.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{kd}}+1$.
8: $\mathbf{b}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{nrhs}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $r$ matrix of right-hand sides $B$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, the $n$ by $r$ solution matrix $X$.
9: $\mathbf{pdb}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
10: $\mathbf{rcond}$double * Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
11: $\mathbf{errbnd}$double * Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that ${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array b and $x$ is the corresponding column of the exact solution $X$. If rcond is less than machine precision, errbnd is returned as unity.
12: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
The Integer allocatable memory required is n, and the double allocatable memory required is $3×{\mathbf{n}}$. Allocation failed before the solution could be computed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{kd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{kd}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}>0$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$ and ${\mathbf{kd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{kd}}+1$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_POS_DEF
The principal minor of order $〈\mathit{\text{value}}〉$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.
NE_RCOND
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.

## 7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b,$
where
 $E1=Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. f04bfc uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f04bfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f04bfc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The band storage schemes for the array ab are identical to the storage schemesfor symmetric and Hermitian band matrices in Chapter F07. See Section 3.4.4 in the F07 Chapter Introduction for details of the storage schemes and an illustrated example.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ then the elements of the stored upper triangular part of $A$ are overwritten by the corresponding elements of the upper triangular matrix $U$. Similarly, if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$ then the elements of the stored lower triangular part of $A$ are overwritten by the corresponding elements of the lower triangular matrix $L$.
Assuming that $n\gg k$, the total number of floating-point operations required to solve the equations $AX=B$ is approximately ${n\left(k+1\right)}^{2}$ for the factorization and $4nkr$ for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The complex analogue of f04bfc is f04cfc.

## 10Example

This example solves the equations
 $AX=B,$
where $A$ is the symmetric positive definite band matrix
 $A= 5.49 2.68 0 0 2.68 5.63 -2.39 0 0 -2.39 2.60 -2.22 0 0 -2.22 5.17 and B= 22.09 5.10 9.31 30.81 -5.24 -25.82 11.83 22.90 .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.

### 10.1Program Text

Program Text (f04bfce.c)

### 10.2Program Data

Program Data (f04bfce.d)

### 10.3Program Results

Program Results (f04bfce.r)