F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07HBF (DPBSVX)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07HBF (DPBSVX) uses the Cholesky factorization
 $A=UTU or A=LLT$
to compute the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ symmetric positive definite band matrix of bandwidth $\left(2{k}_{d}+1\right)$ and $X$ and $B$ are $n$ by $r$ matrices. Error bounds on the solution and a condition estimate are also provided.

## 2  Specification

 SUBROUTINE F07HBF ( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
 INTEGER N, KD, NRHS, LDAB, LDAFB, LDB, LDX, IWORK(N), INFO REAL (KIND=nag_wp) AB(LDAB,*), AFB(LDAFB,*), S(*), B(LDB,*), X(LDX,*), RCOND, FERR(NRHS), BERR(NRHS), WORK(3*N) CHARACTER(1) FACT, UPLO, EQUED
The routine may be called by its LAPACK name dpbsvx.

## 3  Description

F07HBF (DPBSVX) performs the following steps:
1. If ${\mathbf{FACT}}=\text{'E'}$, real diagonal scaling factors, ${D}_{S}$, are computed to equilibrate the system:
 $DS A DS DS-1 X = DS B .$
Whether or not the system will be equilibrated depends on the scaling of the matrix $A$, but if equilibration is used, $A$ is overwritten by ${D}_{S}A{D}_{S}$ and $B$ by ${D}_{S}B$.
2. If ${\mathbf{FACT}}=\text{'N'}$ or $\text{'E'}$, the Cholesky decomposition is used to factor the matrix $A$ (after equilibration if ${\mathbf{FACT}}=\text{'E'}$) as $A={U}^{\mathrm{T}}U$ if ${\mathbf{UPLO}}=\text{'U'}$ or $A=L{L}^{\mathrm{T}}$ if ${\mathbf{UPLO}}=\text{'L'}$, where $U$ is an upper triangular matrix and $L$ is a lower triangular matrix.
3. If the leading $i$ by $i$ principal minor of $A$ is not positive definite, then the routine returns with ${\mathbf{INFO}}=i$. 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, ${\mathbf{INFO}}=\mathbf{N}+{\mathbf{1}}$ is returned as a warning, but the routine still goes on to solve for $X$ and compute error bounds as described below.
4. The system of equations is solved for $X$ using the factored form of $A$.
5. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
6. If equilibration was used, the matrix $X$ is premultiplied by ${D}_{S}$ so that it solves the original system before equilibration.

## 4  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

## 5  Parameters

1:     FACT – CHARACTER(1)Input
On entry: specifies whether or not the factorized form of the matrix $A$ is supplied on entry, and if not, whether the matrix $A$ should be equilibrated before it is factorized.
${\mathbf{FACT}}=\text{'F'}$
AFB contains the factorized form of $A$. If ${\mathbf{EQUED}}=\text{'Y'}$, the matrix $A$ has been equilibrated with scaling factors given by S. AB and AFB will not be modified.
${\mathbf{FACT}}=\text{'N'}$
The matrix $A$ will be copied to AFB and factorized.
${\mathbf{FACT}}=\text{'E'}$
The matrix $A$ will be equilibrated if necessary, then copied to AFB and factorized.
Constraint: ${\mathbf{FACT}}=\text{'F'}$, $\text{'N'}$ or $\text{'E'}$.
2:     UPLO – CHARACTER(1)Input
On entry: if ${\mathbf{UPLO}}=\text{'U'}$, the upper triangle of $A$ is stored.
If ${\mathbf{UPLO}}=\text{'L'}$, the lower triangle of $A$ is stored.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
3:     N – INTEGERInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
4:     KD – INTEGERInput
On entry: ${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{UPLO}}=\text{'U'}$, or the number of subdiagonals if ${\mathbf{UPLO}}=\text{'L'}$.
Constraint: ${\mathbf{KD}}\ge 0$.
5:     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:     AB(LDAB,$*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AB must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the upper or lower triangle of the symmetric band matrix $A$, except if ${\mathbf{FACT}}=\text{'F'}$ and ${\mathbf{EQUED}}=\text{'Y'}$, in which case AB must contain the equilibrated matrix ${D}_{S}A{D}_{S}$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{UPLO}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
On exit: if ${\mathbf{FACT}}=\text{'E'}$ and ${\mathbf{EQUED}}=\text{'Y'}$, AB is overwritten by ${D}_{S}A{D}_{S}$.
7:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07HBF (DPBSVX) is called.
Constraint: ${\mathbf{LDAB}}\ge {\mathbf{KD}}+1$.
8:     AFB(LDAFB,$*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AFB must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: if ${\mathbf{FACT}}=\text{'F'}$, AFB contains the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$ of the band matrix $A$, in the same storage format as $A$. If ${\mathbf{EQUED}}=\text{'Y'}$, AFB is the factorized form of the equilibrated matrix $A$.
On exit: if ${\mathbf{FACT}}=\text{'N'}$, AFB returns the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$.
If ${\mathbf{FACT}}=\text{'E'}$, AFB returns the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$ of the equilibrated matrix $A$ (see the description of AB for the form of the equilibrated matrix).
9:     LDAFB – INTEGERInput
On entry: the first dimension of the array AFB as declared in the (sub)program from which F07HBF (DPBSVX) is called.
Constraint: ${\mathbf{LDAFB}}\ge {\mathbf{KD}}+1$.
10:   EQUED – CHARACTER(1)Input/Output
On entry: if ${\mathbf{FACT}}=\text{'N'}$ or $\text{'E'}$, EQUED need not be set.
If ${\mathbf{FACT}}=\text{'F'}$, EQUED must specify the form of the equilibration that was performed as follows:
• if ${\mathbf{EQUED}}=\text{'N'}$, no equilibration;
• if ${\mathbf{EQUED}}=\text{'Y'}$, equilibration was performed, i.e., $A$ has been replaced by ${D}_{S}A{D}_{S}$.
On exit: if ${\mathbf{FACT}}=\text{'F'}$, EQUED is unchanged from entry.
Otherwise, if no constraints are violated, EQUED specifies the form of the equilibration that was performed as specified above.
Constraint: if ${\mathbf{FACT}}=\text{'F'}$, ${\mathbf{EQUED}}=\text{'N'}$ or $\text{'Y'}$.
11:   S($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array S must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: if ${\mathbf{FACT}}=\text{'N'}$ or $\text{'E'}$, S need not be set.
If ${\mathbf{FACT}}=\text{'F'}$ and ${\mathbf{EQUED}}=\text{'Y'}$, S must contain the scale factors, ${D}_{S}$, for $A$; each element of S must be positive.
On exit: if ${\mathbf{FACT}}=\text{'F'}$, S is unchanged from entry.
Otherwise, if no constraints are violated and ${\mathbf{EQUED}}=\text{'Y'}$, S contains the scale factors, ${D}_{S}$, for $A$; each element of S is positive.
12:   B(LDB,$*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: if ${\mathbf{EQUED}}=\text{'N'}$, B is not modified.
If ${\mathbf{EQUED}}=\text{'Y'}$, B is overwritten by ${D}_{S}B$.
13:   LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07HBF (DPBSVX) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
14:   X(LDX,$*$) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array X must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$ to the original system of equations. Note that the arrays $A$ and $B$ are modified on exit if ${\mathbf{EQUED}}=\text{'Y'}$, and the solution to the equilibrated system is ${D}_{S}^{-1}X$.
15:   LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F07HBF (DPBSVX) is called.
Constraint: ${\mathbf{LDX}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
16:   RCOND – REAL (KIND=nag_wp)Output
On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix $A$ (after equilibration if that is performed), computed as ${\mathbf{RCOND}}=1.0/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
17:   FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, an estimate of the forward error bound for each computed solution vector, such that ${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{x}_{j}‖}_{\infty }\le {\mathbf{FERR}}\left(j\right)$ 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.
18:   BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, 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).
19:   WORK($3×{\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
20:   IWORK(N) – INTEGER arrayWorkspace
21:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If ${\mathbf{INFO}}=i$ and $i\le {\mathbf{N}}$, the leading minor of order $i$ 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.
${\mathbf{INFO}}={\mathbf{N}}+1$
The triangular matrix $U$ (or $L$) 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.

## 7  Accuracy

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
• if ${\mathbf{UPLO}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|{U}^{\mathrm{T}}\right|\left|U\right|$;
• if ${\mathbf{UPLO}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|L\right|\left|{L}^{\mathrm{T}}\right|$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. See Section 10.1 of Higham (2002) for further details.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ 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 ${\mathbf{BERR}}\left(j\right)$ and a bound on ${‖x-\stackrel{^}{x}‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }$ is returned in ${\mathbf{FERR}}\left(j\right)$. See Section 4.4 of Anderson et al. (1999) for further details.

When $n\gg k$, the factorization of $A$ requires approximately $n{\left(k+1\right)}^{2}$ floating point operations, where $k$ is the number of superdiagonals.
For each right-hand side, computation of the backward error involves a minimum of $8nk$ floating point operations. Each step of iterative refinement involves an additional $12nk$ operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $4nk$ operations.
The complex analogue of this routine is F07HPF (ZPBSVX).

## 9  Example

This example solves the equations
 $AX=B ,$
where $A$ is the symmetric positive definite band matrix
 $A = 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17$
and
 $B = 22.09 5.10 9.31 30.81 -5.24 -25.82 11.83 22.90 .$
Error estimates for the solutions, information on equilibration and an estimate of the reciprocal of the condition number of the scaled matrix $A$ are also output.

### 9.1  Program Text

Program Text (f07hbfe.f90)

### 9.2  Program Data

Program Data (f07hbfe.d)

### 9.3  Program Results

Program Results (f07hbfe.r)