# NAG CL Interfacef04bcc (real_​tridiag_​solve)

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## 1Purpose

f04bcc computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n×n$ tridiagonal matrix and $X$ and $B$ are $n×r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

## 2Specification

 #include
 void f04bcc (Nag_OrderType order, Integer n, Integer nrhs, double dl[], double d[], double du[], double du2[], Integer ipiv[], double b[], Integer pdb, double *rcond, double *errbnd, NagError *fail)
The function may be called by the names: f04bcc, nag_linsys_real_tridiag_solve or nag_real_tridiag_lin_solve.

## 3Description

The $LU$ decomposition with partial pivoting and row interchanges is used to factor $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element, and $U$ is an upper triangular band matrix with two superdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.
Note that the equations ${A}^{\mathrm{T}}X=B$ may be solved by interchanging the order of the arguments du and dl.

## 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: $\mathbf{n}$Integer Input
On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{nrhs}$Integer Input
On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
4: $\mathbf{dl}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array dl must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, dl is overwritten by the $\left(n-1\right)$ multipliers that define the matrix $L$ from the $LU$ factorization of $A$.
5: $\mathbf{d}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: must contain the $n$ diagonal elements of the matrix $A$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, d is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
6: $\mathbf{du}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array du must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ superdiagonal elements of the matrix $A$
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, du is overwritten by the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
7: $\mathbf{du2}\left[{\mathbf{n}}-2\right]$double Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, du2 returns the $\left(n-2\right)$ elements of the second superdiagonal of $U$.
8: $\mathbf{ipiv}\left[{\mathbf{n}}\right]$Integer Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left[i-1\right]$. ${\mathbf{ipiv}}\left[i-1\right]$ will always be either $i$ or $\left(i+1\right)$; ${\mathbf{ipiv}}\left[i-1\right]=i$ indicates a row interchange was not required.
9: $\mathbf{b}\left[\mathit{dim}\right]$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 $n×r$ matrix of right-hand sides $B$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, the $n×r$ solution matrix $X$.
10: $\mathbf{pdb}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
11: $\mathbf{rcond}$double * Output
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
12: $\mathbf{errbnd}$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.
13: $\mathbf{fail}$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 $2×{\mathbf{n}}$. In this case the factorization and the solution $X$ have been computed, but rcond and errbnd have not been 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, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{nrhs}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
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_RCOND
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
NE_SINGULAR
Diagonal element $⟨\mathit{\text{value}}⟩$ of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.

## 7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $(A+E) x^=b,$
where
 $‖E‖1 = O(ε) ‖A‖1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. f04bcc 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

f04bcc 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 total number of floating-point operations required to solve the equations $AX=B$ is proportional to $nr$. 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 f04bcc is f04ccc.

## 10Example

This example solves the equations
 $AX=B,$
where $A$ is the tridiagonal matrix
 $A= ( 3.0 2.1 0 0 0 3.4 2.3 -1.0 0 0 0 3.6 -5.0 1.9 0 0 0 7.0 -0.9 8.0 0 0 0 -6.0 7.1 ) and B= ( 2.7 6.6 -0.5 10.8 2.6 -3.2 0.6 -11.2 2.7 19.1 ) .$
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 (f04bcce.c)

### 10.2Program Data

Program Data (f04bcce.d)

### 10.3Program Results

Program Results (f04bcce.r)