f16 Chapter Contents
f16 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dtbsv (f16pkc)

## 1  Purpose

nag_dtbsv (f16pkc) solves a system of equations given as a real triangular band matrix.

## 2  Specification

 #include #include
 void nag_dtbsv (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Nag_DiagType diag, Integer n, Integer k, double alpha, const double ab[], Integer pdab, double x[], Integer incx, NagError *fail)

## 3  Description

nag_dtbsv (f16pkc) performs one of the matrix-vector operations
 $x←αA-1x or x←αA-Tx,$
where $A$ is an $n$ by $n$ real triangular band matrix with $k$ subdiagonals or superdiagonals, $x$ is an $n$-element real vector and $\alpha$ is a real scalar. ${A}^{-\mathrm{T}}$ denotes ${\left({A}^{\mathrm{T}}\right)}^{-1}$ or equivalently ${\left({A}^{-1}\right)}^{\mathrm{T}}$.
No test for singularity or near-singularity of $A$ is included in this function. Such tests must be performed before calling this function.

## 4  References

The BLAS Technical Forum Standard (2001) http://www.netlib.org/blas/blast-forum

## 5  Arguments

1:     orderNag_OrderTypeInput
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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: specifies whether $A$ is upper or lower triangular.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
$A$ is upper triangular.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
$A$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:     transNag_TransTypeInput
On entry: specifies the operation to be performed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$x←\alpha {A}^{-1}x$.
${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$
$x←\alpha {A}^{-\mathrm{T}}x$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, $\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
4:     diagNag_DiagTypeInput
On entry: specifies whether $A$ has nonunit or unit diagonal elements.
${\mathbf{diag}}=\mathrm{Nag_NonUnitDiag}$
The diagonal elements are stored explicitly.
${\mathbf{diag}}=\mathrm{Nag_UnitDiag}$
The diagonal elements are assumed to be $1$ and are not referenced.
Constraint: ${\mathbf{diag}}=\mathrm{Nag_NonUnitDiag}$ or $\mathrm{Nag_UnitDiag}$.
5:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     kIntegerInput
On entry: $k$, the number of subdiagonals or superdiagonals of the matrix $A$.
Constraint: ${\mathbf{k}}\ge 0$.
On entry: the scalar $\alpha$.
8:     ab[$\mathit{dim}$]const doubleInput
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: the $n$ by $n$ triangular band matrix $A$.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of ${A}_{ij}$, depends on the order and uplo arguments as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[k+i-j+\left(j-1\right)×{\mathbf{pdab}}\right]$, for $j=1,\dots ,n$ and $i=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-k\right),\dots ,j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[i-j+\left(j-1\right)×{\mathbf{pdab}}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+k\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[j-i+\left(i-1\right)×{\mathbf{pdab}}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+k\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[k+j-i+\left(i-1\right)×{\mathbf{pdab}}\right]$, for $i=1,\dots ,n$ and $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-k\right),\dots ,i$.
If ${\mathbf{diag}}=\mathrm{Nag_UnitDiag}$, the diagonal elements of $\mathrm{AB}$ are assumed to be $1$, and are not referenced.
9:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array ab.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{k}}+1$.
10:   x[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)\left|{\mathbf{incx}}\right|\right)$.
On entry: the right-hand side vector $b$.
On exit: the solution vector $x$.
11:   incxIntegerInput
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}\ne 0$.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{incx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{incx}}\ne 0$.
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{k}}+1$.

## 7  Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of The BLAS Technical Forum Standard (2001)).

None.

## 9  Example

This example solves the real triangular band system of linear equations $Ax=y$, where $A$ is the $4$ by $4$ triangular matrix given with one subdiagonal given by
 $A = -4.16 -2.25 4.78 5.86 6.32 -4.82 0.16$
and where
 $y = -16.64,-13.78,13.10,-14.14T .$
$A$ is stored in array ab using banded storage format and $y$ is stored in array x. nag_dtbsv (f16pkc) returns the solution in x.

### 9.1  Program Text

Program Text (f16pkce.c)

### 9.2  Program Data

Program Data (f16pkce.d)

### 9.3  Program Results

Program Results (f16pkce.r)