# NAG C Library Function Document

## 1Purpose

nag_dgtsvx (f07cbc) uses the $LU$ factorization to compute the solution to a real system of linear equations
 $AX=B or ATX=B ,$
where $A$ is a tridiagonal matrix of order $n$ and $X$ and $B$ are $n$ by $r$ matrices. Error bounds on the solution and a condition estimate are also provided.

## 2Specification

 #include #include
 void nag_dgtsvx (Nag_OrderType order, Nag_FactoredFormType fact, Nag_TransType trans, Integer n, Integer nrhs, const double dl[], const double d[], const double du[], double dlf[], double df[], double duf[], double du2[], Integer ipiv[], const double b[], Integer pdb, double x[], Integer pdx, double *rcond, double ferr[], double berr[], NagError *fail)

## 3Description

nag_dgtsvx (f07cbc) performs the following steps:
 1 If ${\mathbf{fact}}=\mathrm{Nag_NotFactored}$, the $LU$ decomposition is used to factor the matrix $A$ as $A=LU$, where $L$ is a product of permutation and unit lower bidiagonal matrices and $U$ is upper triangular with nonzeros in only the main diagonal and first two superdiagonals. 2 If some ${u}_{ii}=0$, so that $U$ is exactly singular, then the function returns with ${\mathbf{fail}}\mathbf{.}\mathbf{errnum}=i$. Otherwise, the factored form of $A$ is used to estimate the condition number of the matrix $A$. If the reciprocal of the condition number is less than machine precision, ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_SINGULAR_WP is returned as a warning, but the function still goes on to solve for $X$ and compute error bounds as described below. 3 The system of equations is solved for $X$ using the factored form of $A$. 4 Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

## 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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## 5Arguments

1:    $\mathbf{order}$Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{fact}$Nag_FactoredFormTypeInput
On entry: specifies whether or not the factorized form of the matrix $A$ has been supplied.
${\mathbf{fact}}=\mathrm{Nag_Factored}$
dlf, df, duf, du2 and ipiv contain the factorized form of the matrix $A$. dlf, df, duf, du2 and ipiv will not be modified.
${\mathbf{fact}}=\mathrm{Nag_NotFactored}$
The matrix $A$ will be copied to dlf, df and duf and factorized.
Constraint: ${\mathbf{fact}}=\mathrm{Nag_Factored}$ or $\mathrm{Nag_NotFactored}$.
3:    $\mathbf{trans}$Nag_TransTypeInput
On entry: specifies the form of the system of equations.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$AX=B$ (No transpose).
${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$
${A}^{\mathrm{T}}X=B$ (Transpose).
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, $\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:    $\mathbf{nrhs}$IntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
6:    $\mathbf{dl}\left[\mathit{dim}\right]$const doubleInput
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: the $\left(n-1\right)$ subdiagonal elements of $A$.
7:    $\mathbf{d}\left[\mathit{dim}\right]$const doubleInput
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: the $n$ diagonal elements of $A$.
8:    $\mathbf{du}\left[\mathit{dim}\right]$const doubleInput
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: the $\left(n-1\right)$ superdiagonal elements of $A$.
9:    $\mathbf{dlf}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array dlf must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: if ${\mathbf{fact}}=\mathrm{Nag_Factored}$, dlf contains the $\left(n-1\right)$ multipliers that define the matrix $L$ from the $LU$ factorization of $A$.
On exit: if ${\mathbf{fact}}=\mathrm{Nag_NotFactored}$, dlf contains the $\left(n-1\right)$ multipliers that define the matrix $L$ from the $LU$ factorization of $A$.
10:  $\mathbf{df}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array df must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: if ${\mathbf{fact}}=\mathrm{Nag_Factored}$, df contains the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
On exit: if ${\mathbf{fact}}=\mathrm{Nag_NotFactored}$, df contains the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
11:  $\mathbf{duf}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array duf must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: if ${\mathbf{fact}}=\mathrm{Nag_Factored}$, duf contains the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
On exit: if ${\mathbf{fact}}=\mathrm{Nag_NotFactored}$, duf contains the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
12:  $\mathbf{du2}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array du2 must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-2\right)$.
On entry: if ${\mathbf{fact}}=\mathrm{Nag_Factored}$, du2 contains the ($n-2$) elements of the second superdiagonal of $U$.
On exit: if ${\mathbf{fact}}=\mathrm{Nag_NotFactored}$, du2 contains the ($n-2$) elements of the second superdiagonal of $U$.
13:  $\mathbf{ipiv}\left[\mathit{dim}\right]$IntegerInput/Output
Note: the dimension, dim, of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: if ${\mathbf{fact}}=\mathrm{Nag_Factored}$, ipiv contains the pivot indices from the $LU$ factorization of $A$.
On exit: if ${\mathbf{fact}}=\mathrm{Nag_NotFactored}$, ipiv contains the pivot indices from the $LU$ factorization of $A$; 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 $i+1$; ${\mathbf{ipiv}}\left[i-1\right]=i$ indicates a row interchange was not required.
14:  $\mathbf{b}\left[\mathit{dim}\right]$const doubleInput
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 $r$ right-hand side matrix $B$.
15:  $\mathbf{pdb}$IntegerInput
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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
16:  $\mathbf{x}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdx}}×{\mathbf{nrhs}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdx}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $X$ is stored in
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_SINGULAR_WP, the $n$ by $r$ solution matrix $X$.
17:  $\mathbf{pdx}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
18:  $\mathbf{rcond}$double *Output
On exit: the estimate of the reciprocal condition number of the matrix $A$. If ${\mathbf{rcond}}=0.0$, the matrix may be exactly singular. This condition is indicated by ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_SINGULAR. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_SINGULAR_WP.
19:  $\mathbf{ferr}\left[{\mathbf{nrhs}}\right]$doubleOutput
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_SINGULAR_WP, an estimate of the forward error bound for each computed solution vector, such that ${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{x}_{j}‖}_{\infty }\le {\mathbf{ferr}}\left[j-1\right]$ where ${\stackrel{^}{x}}_{j}$ is the $j$th column of the computed solution returned in the array x and ${x}_{j}$ is the corresponding column of the exact solution $X$. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
20:  $\mathbf{berr}\left[{\mathbf{nrhs}}\right]$doubleOutput
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_SINGULAR_WP, an estimate of the component-wise relative backward error of each computed solution vector ${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\stackrel{^}{x}}_{j}$ an exact solution).
21:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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: ${\mathbf{pdb}}>0$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}>0$.
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: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR
Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. ${\mathbf{rcond}}=0.0$ is returned.
Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero. The factorization has not been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. ${\mathbf{rcond}}=0.0$ is returned.
NE_SINGULAR_WP
$U$ is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

## 7Accuracy

For each right-hand side vector $b$, the computed solution $\stackrel{^}{x}$ is the exact solution of a perturbed system of equations $\left(A+E\right)\stackrel{^}{x}=b$, where
 $E ≤ c n ε L U ,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. See Section 9.3 of Higham (2002) for further details.
If $x$ is the true solution, then the computed solution $\stackrel{^}{x}$ satisfies a forward error bound of the form
 $x-x^∞ x^∞ ≤ wc condA,x^,b$
where $\mathrm{cond}\left(A,\stackrel{^}{x},b\right)={‖\left|{A}^{-1}\right|\left(\left|A\right|\left|\stackrel{^}{x}\right|+\left|b\right|\right)‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. If $\stackrel{^}{x}$ is the $j$th column of $X$, then ${w}_{c}$ is returned in ${\mathbf{berr}}\left[j-1\right]$ and a bound on ${‖x-\stackrel{^}{x}‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }$ is returned in ${\mathbf{ferr}}\left[j-1\right]$. See Section 4.4 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

nag_dgtsvx (f07cbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgtsvx (f07cbc) 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. The solution is then refined, and the errors estimated, using iterative refinement.
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 this function is nag_zgtsvx (f07cpc).

## 10Example

This example solves the equations
 $AX=B ,$
where $A$ is the tridiagonal matrix
 $A = 3.0 2.1 0.0 0.0 0.0 3.4 2.3 -1.0 0.0 0.0 0.0 3.6 -5.0 1.9 0.0 0.0 0.0 7.0 -0.9 8.0 0.0 0.0 0.0 -6.0 7.1$
and
 $B = 2.7 6.6 -0.5 10.8 2.6 -3.2 0.6 -11.2 2.7 19.1 .$
Estimates for the backward errors, forward errors and condition number are also output.

### 10.1Program Text

Program Text (f07cbce.c)

### 10.2Program Data

Program Data (f07cbce.d)

### 10.3Program Results

Program Results (f07cbce.r)