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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgtsvx (f07cb)

## Purpose

nag_lapack_dgtsvx (f07cb) uses the LU$LU$ factorization to compute the solution to a real system of linear equations
 AX = B   or   ATX = B , $AX=B or ATX=B ,$
where A$A$ is a tridiagonal matrix of order n$n$ and X$X$ and B$B$ are n$n$ by r$r$ matrices. Error bounds on the solution and a condition estimate are also provided.

## Syntax

[dlf, df, duf, du2, ipiv, x, rcond, ferr, berr, info] = f07cb(fact, trans, dl, d, du, dlf, df, duf, du2, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[dlf, df, duf, du2, ipiv, x, rcond, ferr, berr, info] = nag_lapack_dgtsvx(fact, trans, dl, d, du, dlf, df, duf, du2, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dgtsvx (f07cb) performs the following steps:
1. If fact = 'N'${\mathbf{fact}}=\text{'N'}$, the LU$LU$ decomposition is used to factor the matrix A$A$ as A = LU$A=LU$, where L$L$ is a product of permutation and unit lower bidiagonal matrices and U$U$ is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
2. If some uii = 0${u}_{ii}=0$, so that U$U$ is exactly singular, then the function returns with info = i${\mathbf{info}}=i$. Otherwise, the factored form of A$A$ is used to estimate the condition number of the matrix A$A$. If the reciprocal of the condition number is less than machine precision, infon + 1${\mathbf{info}}\ge {\mathbf{n}}+1$ is returned as a warning, but the function still goes on to solve for X$X$ and compute error bounds as described below.
3. The system of equations is solved for X$X$ using the factored form of A$A$.
4. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

## References

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

## Parameters

### Compulsory Input Parameters

1:     fact – string (length ≥ 1)
Specifies whether or not the factorized form of the matrix A$A$ has been supplied.
fact = 'F'${\mathbf{fact}}=\text{'F'}$
dlf, df, duf, du2 and ipiv contain the factorized form of the matrix A$A$. dlf, df, duf, du2 and ipiv will not be modified.
fact = 'N'${\mathbf{fact}}=\text{'N'}$
The matrix A$A$ will be copied to dlf, df and duf and factorized.
Constraint: fact = 'F'${\mathbf{fact}}=\text{'F'}$ or 'N'$\text{'N'}$.
2:     trans – string (length ≥ 1)
Specifies the form of the system of equations.
trans = 'N'${\mathbf{trans}}=\text{'N'}$
AX = B$AX=B$ (No transpose).
trans = 'T'${\mathbf{trans}}=\text{'T'}$ or 'C'$\text{'C'}$
ATX = B${A}^{\mathrm{T}}X=B$ (Transpose).
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$, 'T'$\text{'T'}$ or 'C'$\text{'C'}$.
3:     dl( : $:$) – double array
Note: the dimension of the array dl must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
The (n1)$\left(n-1\right)$ subdiagonal elements of A$A$.
4:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ diagonal elements of A$A$.
5:     du( : $:$) – double array
Note: the dimension of the array du must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
The (n1)$\left(n-1\right)$ superdiagonal elements of A$A$.
6:     dlf( : $:$) – double array
Note: the dimension of the array dlf must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If fact = 'F'${\mathbf{fact}}=\text{'F'}$, dlf contains the (n1)$\left(n-1\right)$ multipliers that define the matrix L$L$ from the LU$LU$ factorization of A$A$.
7:     df( : $:$) – double array
Note: the dimension of the array df must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If fact = 'F'${\mathbf{fact}}=\text{'F'}$, df contains the n$n$ diagonal elements of the upper triangular matrix U$U$ from the LU$LU$ factorization of A$A$.
8:     duf( : $:$) – double array
Note: the dimension of the array duf must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If fact = 'F'${\mathbf{fact}}=\text{'F'}$, duf contains the (n1)$\left(n-1\right)$ elements of the first superdiagonal of U$U$.
9:     du2( : $:$) – double array
Note: the dimension of the array du2 must be at least max (1,n2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-2\right)$.
If fact = 'F'${\mathbf{fact}}=\text{'F'}$, du2 contains the (n2$n-2$) elements of the second superdiagonal of U$U$.
10:   ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If fact = 'F'${\mathbf{fact}}=\text{'F'}$, ipiv contains the pivot indices from the LU$LU$ factorization of A$A$.
11:   b(ldb, : $:$) – double array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ right-hand side matrix B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b The dimension of the arrays d, df, ipiv.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
r$r$, the number of right-hand sides, i.e., the number of columns of the matrix B$B$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

ldb ldx work iwork

### Output Parameters

1:     dlf( : $:$) – double array
Note: the dimension of the array dlf must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If fact = 'N'${\mathbf{fact}}=\text{'N'}$, dlf contains the (n1)$\left(n-1\right)$ multipliers that define the matrix L$L$ from the LU$LU$ factorization of A$A$.
2:     df( : $:$) – double array
Note: the dimension of the array df must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If fact = 'N'${\mathbf{fact}}=\text{'N'}$, df contains the n$n$ diagonal elements of the upper triangular matrix U$U$ from the LU$LU$ factorization of A$A$.
3:     duf( : $:$) – double array
Note: the dimension of the array duf must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If fact = 'N'${\mathbf{fact}}=\text{'N'}$, duf contains the (n1)$\left(n-1\right)$ elements of the first superdiagonal of U$U$.
4:     du2( : $:$) – double array
Note: the dimension of the array du2 must be at least max (1,n2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-2\right)$.
If fact = 'N'${\mathbf{fact}}=\text{'N'}$, du2 contains the (n2$n-2$) elements of the second superdiagonal of U$U$.
5:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If fact = 'N'${\mathbf{fact}}=\text{'N'}$, ipiv contains the pivot indices from the LU$LU$ factorization of A$A$; row i$i$ of the matrix was interchanged with row ipiv(i)${\mathbf{ipiv}}\left(i\right)$. ipiv(i)${\mathbf{ipiv}}\left(i\right)$ will always be either i$i$ or i + 1$i+1$; ipiv(i) = i${\mathbf{ipiv}}\left(i\right)=i$ indicates a row interchange was not required.
6:     x(ldx, : $:$) – double array
The first dimension of the array x will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldxmax (1,n)$\mathit{ldx}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{INFO}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, the n$n$ by r$r$ solution matrix X$X$.
7:     rcond – double scalar
The estimate of the reciprocal condition number of the matrix A$A$. If rcond = 0.0${\mathbf{rcond}}=0.0$, the matrix may be exactly singular. This condition is indicated by INFO > 0andINFOn${\mathbf{INFO}}>{\mathbf{0}} \text{and} {\mathbf{INFO}}\le \mathbf{n}$. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by infon + 1${\mathbf{info}}\ge {\mathbf{n}}+1$.
8:     ferr(nrhs_p) – double array
If ${\mathbf{INFO}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for each computed solution vector, such that jxj / xjferr(j)${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{x}_{j}‖}_{\infty }\le {\mathbf{ferr}}\left(j\right)$ where j${\stackrel{^}{x}}_{j}$ is the j$j$th column of the computed solution returned in the array x and xj${x}_{j}$ is the corresponding column of the exact solution X$X$. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
9:     berr(nrhs_p) – double array
If ${\mathbf{INFO}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, an estimate of the component-wise relative backward error of each computed solution vector j${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of A$A$ or B$B$ that makes j${\stackrel{^}{x}}_{j}$ an exact solution).
10:   info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: fact, 2: trans, 3: n, 4: nrhs_p, 5: dl, 6: d, 7: du, 8: dlf, 9: df, 10: duf, 11: du2, 12: ipiv, 13: b, 14: ldb, 15: x, 16: ldx, 17: rcond, 18: ferr, 19: berr, 20: work, 21: iwork, 22: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0andINFON${\mathbf{INFO}}>0 \text{and} {\mathbf{INFO}}\le {\mathbf{N}}$
If info = i${\mathbf{info}}=i$, u(i,i)$u\left(i,i\right)$ is exactly zero. The factorization has not been completed unless i = n$i={\mathbf{n}}$, but the factor U$U$ is exactly singular, so the solution and error bounds could not be computed. rcond = 0.0${\mathbf{rcond}}=0.0$ is returned.
W INFO = N + 1${\mathbf{INFO}}={\mathbf{N}}+1$
The triangular matrix U$U$ 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.

## Accuracy

For each right-hand side vector b$b$, the computed solution $\stackrel{^}{x}$ is the exact solution of a perturbed system of equations (A + E) = b$\left(A+E\right)\stackrel{^}{x}=b$, where
 |E| ≤ c (n) ε |L| |U| , $|E| ≤ c (n) ε |L| |U| ,$
c(n)$c\left(n\right)$ is a modest linear function of n$n$, and ε$\epsilon$ is the machine precision. See Section 9.3 of Higham (2002) for further details.
If x$x$ is the true solution, then the computed solution $\stackrel{^}{x}$ satisfies a forward error bound of the form
 ( ‖x − x̂‖∞ )/( ‖x̂‖∞ ) ≤ wc cond(A,x̂,b) $‖x-x^‖∞ ‖x^‖∞ ≤ wc cond(A,x^,b)$
where cond(A,,b) = |A1|(|A||| + |b|) / cond(A) = |A1||A|κ (A)$\mathrm{cond}\left(A,\stackrel{^}{x},b\right)={‖|{A}^{-1}|\left(|A||\stackrel{^}{x}|+|b|\right)‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. If $\stackrel{^}{x}$ is the j $j$th column of X $X$, then wc ${w}_{c}$ is returned in berr(j) ${\mathbf{berr}}\left(j\right)$ and a bound on x / ${‖x-\stackrel{^}{x}‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }$ is returned in ferr(j) ${\mathbf{ferr}}\left(j\right)$. See Section 4.4 of Anderson et al. (1999) for further details.

The total number of floating point operations required to solve the equations AX = B $AX=B$ is proportional to nr $nr$.
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_lapack_zgtsvx (f07cp).

## Example

```function nag_lapack_dgtsvx_example
fact = 'No factors';
trans = 'No transpose';
dl = [3.4;
3.6;
7;
-6];
d = [3;
2.3;
-5;
-0.9;
7.1];
du = [2.1;
-1;
1.9;
8];
dlf = zeros(4, 1);
df  = zeros(5, 1);
duf = zeros(4, 1);
du2 = zeros(3, 1);
ipiv = [int64(8209208);8;-1233199012;-1233198944;24641422];
b = [2.7, 6.6;
-0.5, 10.8;
2.6, -3.2;
0.6, -11.2;
2.7, 19.1];
[dlfOut, dfOut, dufOut, du2Out, ipivOut, x, rcond, ferr, berr, info] = ...
nag_lapack_dgtsvx(fact, trans, dl, d, du, dlf, df, duf, du2, ipiv, b)
```
```

dlfOut =

0.8824
0.0196
0.1401
-0.0148

dfOut =

3.4000
3.6000
7.0000
-6.0000
-1.0154

dufOut =

2.3000
-5.0000
-0.9000
7.1000

du2Out =

-1.0000
1.9000
8.0000

ipivOut =

2
3
4
5
5

x =

-4.0000    5.0000
7.0000   -4.0000
3.0000   -3.0000
-4.0000   -2.0000
-3.0000    1.0000

rcond =

0.0108

ferr =

1.0e-13 *

0.0940
0.1413

berr =

1.0e-16 *

0.7221
0.5921

info =

0

```
```function f07cb_example
fact = 'No factors';
trans = 'No transpose';
dl = [3.4;
3.6;
7;
-6];
d = [3;
2.3;
-5;
-0.9;
7.1];
du = [2.1;
-1;
1.9;
8];
dlf = zeros(4, 1);
df  = zeros(5, 1);
duf = zeros(4, 1);
du2 = zeros(3, 1);
ipiv = [int64(8209208);8;-1233199012;-1233198944;24641422];
b = [2.7, 6.6;
-0.5, 10.8;
2.6, -3.2;
0.6, -11.2;
2.7, 19.1];
[dlfOut, dfOut, dufOut, du2Out, ipivOut, x, rcond, ferr, berr, info] = ...
f07cb(fact, trans, dl, d, du, dlf, df, duf, du2, ipiv, b)
```
```

dlfOut =

0.8824
0.0196
0.1401
-0.0148

dfOut =

3.4000
3.6000
7.0000
-6.0000
-1.0154

dufOut =

2.3000
-5.0000
-0.9000
7.1000

du2Out =

-1.0000
1.9000
8.0000

ipivOut =

2
3
4
5
5

x =

-4.0000    5.0000
7.0000   -4.0000
3.0000   -3.0000
-4.0000   -2.0000
-3.0000    1.0000

rcond =

0.0108

ferr =

1.0e-13 *

0.0940
0.1413

berr =

1.0e-16 *

0.7221
0.5921

info =

0

```