# NAG CL Interfacef07jbc (dptsvx)

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## 1Purpose

f07jbc uses the factorization
 $A=LDLT$
to compute the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n×n$ symmetric positive definite tridiagonal matrix and $X$ and $B$ are $n×r$ matrices. Error bounds on the solution and a condition estimate are also provided.

## 2Specification

 #include
 void f07jbc (Nag_OrderType order, Nag_FactoredFormType fact, Integer n, Integer nrhs, const double d[], const double e[], double df[], double ef[], const double b[], Integer pdb, double x[], Integer pdx, double *rcond, double ferr[], double berr[], NagError *fail)
The function may be called by the names: f07jbc, nag_lapacklin_dptsvx or nag_dptsvx.

## 3Description

f07jbc performs the following steps:
1. 1.If ${\mathbf{fact}}=\mathrm{Nag_NotFactored}$, the matrix $A$ is factorized as $A=LD{L}^{\mathrm{T}}$, where $L$ is a unit lower bidiagonal matrix and $D$ is diagonal. The factorization can also be regarded as having the form $A={U}^{\mathrm{T}}DU$.
2. 2.If the leading $i×i$ principal minor is not positive definite, then the function returns with ${\mathbf{fail}}\mathbf{.}\mathbf{errnum}=i$ and ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_MAT_NOT_POS_DEF. 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. 3.The system of equations is solved for $X$ using the factored form of $A$.
4. 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 https://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_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: $\mathbf{fact}$Nag_FactoredFormType Input
On entry: specifies whether or not the factorized form of the matrix $A$ has been supplied.
${\mathbf{fact}}=\mathrm{Nag_Factored}$
df and ef contain the factorized form of the matrix $A$. df and ef will not be modified.
${\mathbf{fact}}=\mathrm{Nag_NotFactored}$
The matrix $A$ will be copied to df and ef and factorized.
Constraint: ${\mathbf{fact}}=\mathrm{Nag_Factored}$ or $\mathrm{Nag_NotFactored}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
5: $\mathbf{d}\left[\mathit{dim}\right]$const double Input
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 the tridiagonal matrix $A$.
6: $\mathbf{e}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array e 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 the tridiagonal matrix $A$.
7: $\mathbf{df}\left[\mathit{dim}\right]$double Input/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 must contain the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
On exit: if ${\mathbf{fact}}=\mathrm{Nag_NotFactored}$, df contains the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
8: $\mathbf{ef}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array ef must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: if ${\mathbf{fact}}=\mathrm{Nag_Factored}$, ef must contain the $\left(n-1\right)$ subdiagonal elements of the unit bidiagonal factor $L$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
On exit: if ${\mathbf{fact}}=\mathrm{Nag_NotFactored}$, ef contains the $\left(n-1\right)$ subdiagonal elements of the unit bidiagonal factor $L$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
9: $\mathbf{b}\left[\mathit{dim}\right]$const double Input
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 $n×r$ right-hand side matrix $B$.
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: $\mathbf{x}\left[\mathit{dim}\right]$double Output
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 $n×r$ solution matrix $X$.
12: $\mathbf{pdx}$Integer Input
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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
13: $\mathbf{rcond}$double * Output
On exit: the reciprocal condition number of the matrix $A$. If rcond is less than the machine precision (in particular, if ${\mathbf{rcond}}=0.0$), the matrix is singular to working precision. This condition is indicated by a return code of ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_SINGULAR_WP.
14: $\mathbf{ferr}\left[{\mathbf{nrhs}}\right]$double Output
On exit: the forward error bound for each solution vector ${\stackrel{^}{x}}_{j}$ (the $j$th column of the solution matrix $X$). If ${x}_{j}$ is the true solution corresponding to ${\stackrel{^}{x}}_{j}$, ${\mathbf{ferr}}\left[j-1\right]$ is an estimated upper bound for the magnitude of the largest element in (${\stackrel{^}{x}}_{j}-{x}_{j}$) divided by the magnitude of the largest element in ${\stackrel{^}{x}}_{j}$.
15: $\mathbf{berr}\left[{\mathbf{nrhs}}\right]$double Output
On exit: the component-wise relative backward error of each 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).
16: $\mathbf{fail}$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.
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: ${\mathbf{pdb}}>0$.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MAT_NOT_POS_DEF
The leading minor of order $⟨\mathit{\text{value}}⟩$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed. ${\mathbf{rcond}}=0.0$ is returned.
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_SINGULAR_WP
$D$ 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
 $|E| ≤ c (n) ε |R| |RT| , where ​ R = L D12 ,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. See Section 10.1 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
 $‖x-x^‖∞ ‖x^‖∞ ≤ wc cond(A,x^,b)$
where $\mathrm{cond}\left(A,\stackrel{^}{x},b\right)={‖|{A}^{-1}|\left(|A||\stackrel{^}{x}|+|b|\right)‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. If $\stackrel{^}{x}$ is the $j$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

f07jbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07jbc 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 number of floating-point operations required for the factorization, and for the estimation of the condition number of $A$ is proportional to $n$. The number of floating-point operations required for the solution of the equations, and for the estimation of the forward and backward error is proportional to $nr$, where $r$ is the number of right-hand sides.
The condition estimation is based upon Equation (15.11) of Higham (2002). For further details of the error estimation, see Section 4.4 of Anderson et al. (1999).
The complex analogue of this function is f07jpc.

## 10Example

This example solves the equations
 $AX=B ,$
where $A$ is the symmetric positive definite tridiagonal matrix
 $A = ( 4.0 -2.0 0 0 0 -2.0 10.0 -6.0 0 0 0 -6.0 29.0 15.0 0 0 0 15.0 25.0 8.0 0 0 0 8.0 5.0 )$
and
 $B = ( 6.0 10.0 9.0 4.0 2.0 9.0 14.0 65.0 7.0 23.0 ) .$
Error estimates for the solutions and an estimate of the reciprocal of the condition number of $A$ are also output.

### 10.1Program Text

Program Text (f07jbce.c)

### 10.2Program Data

Program Data (f07jbce.d)

### 10.3Program Results

Program Results (f07jbce.r)