NAG Library Routine Document

1Purpose

f07hpf (zpbsvx) uses the Cholesky factorization
 $A=UHU or A=LLH$
to compute the solution to a complex system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ Hermitian 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.

2Specification

Fortran Interface
 Subroutine f07hpf ( fact, uplo, n, kd, nrhs, ab, ldab, afb, s, b, ldb, x, ldx, ferr, berr, work, info)
 Integer, Intent (In) :: n, kd, nrhs, ldab, ldafb, ldb, ldx Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: s(*) Real (Kind=nag_wp), Intent (Out) :: rcond, ferr(nrhs), berr(nrhs), rwork(n) Complex (Kind=nag_wp), Intent (Inout) :: ab(ldab,*), afb(ldafb,*), b(ldb,*), x(ldx,*) Complex (Kind=nag_wp), Intent (Out) :: work(2*n) Character (1), Intent (In) :: fact, uplo Character (1), Intent (Inout) :: equed
#include nagmk26.h
 void f07hpf_ (const char *fact, const char *uplo, const Integer *n, const Integer *kd, const Integer *nrhs, Complex ab[], const Integer *ldab, Complex afb[], const Integer *ldafb, char *equed, double s[], Complex b[], const Integer *ldb, Complex x[], const Integer *ldx, double *rcond, double ferr[], double berr[], Complex work[], double rwork[], Integer *info, const Charlen length_fact, const Charlen length_uplo, const Charlen length_equed)
The routine may be called by its LAPACK name zpbsvx.

3Description

f07hpf (zpbsvx) 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 $A={U}^{\mathrm{H}}U$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=L{L}^{\mathrm{H}}$ 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.

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:     $\mathbf{fact}$ – 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:     $\mathbf{uplo}$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathbf{kd}$ – 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:     $\mathbf{nrhs}$ – 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:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Complex (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 Hermitian 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:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hpf (zpbsvx) is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
8:     $\mathbf{afb}\left({\mathbf{ldafb}},*\right)$ – Complex (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 $A={U}^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$ 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 $A={U}^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$.
If ${\mathbf{fact}}=\text{'E'}$, afb returns the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$ of the equilibrated matrix $A$ (see the description of ab for the form of the equilibrated matrix).
9:     $\mathbf{ldafb}$ – IntegerInput
On entry: the first dimension of the array afb as declared in the (sub)program from which f07hpf (zpbsvx) is called.
Constraint: ${\mathbf{ldafb}}\ge {\mathbf{kd}}+1$.
10:   $\mathbf{equed}$ – 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:   $\mathbf{s}\left(*\right)$ – 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:   $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (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:   $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07hpf (zpbsvx) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
14:   $\mathbf{x}\left({\mathbf{ldx}},*\right)$ – Complex (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:   $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07hpf (zpbsvx) is called.
Constraint: ${\mathbf{ldx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
16:   $\mathbf{rcond}$ – 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:   $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$ – 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:   $\mathbf{berr}\left({\mathbf{nrhs}}\right)$ – 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:   $\mathbf{work}\left(2×{\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayWorkspace
20:   $\mathbf{rwork}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
21:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0 \text{and} {\mathbf{info}}\le {\mathbf{n}}$
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.
${\mathbf{info}}={\mathbf{n}}+1$
$U$ (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.

7Accuracy

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 ${\mathbf{uplo}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|{U}^{\mathrm{H}}\right|\left|U\right|$;
• if ${\mathbf{uplo}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|L\right|\left|{L}^{\mathrm{H}}\right|$,
$c\left(n\right)$ 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.

8Parallelism and Performance

f07hpf (zpbsvx) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07hpf (zpbsvx) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

When $n\gg k$, the factorization of $A$ requires approximately $4n{\left(k+1\right)}^{2}$ floating-point operations, where $k$ is the number of superdiagonals.
For each right-hand side, computation of the backward error involves a minimum of $32nk$ floating-point operations. Each step of iterative refinement involves an additional $48nk$ 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 $16nk$ operations.
The real analogue of this routine is f07hbf (dpbsvx).

10Example

This example solves the equations
 $AX=B ,$
where $A$ is the Hermitian positive definite band matrix
 $A = 9.39i+0.00 1.08-1.73i 0.00i+0.00 0.00i+0.00 1.08+1.73i 1.69i+0.00 -0.04+0.29i 0.00i+0.00 0.00i+0.00 -0.04-0.29i 2.65i+0.00 -0.33+2.24i 0.00i+0.00 0.00i+0.00 -0.33-2.24i 2.17i+0.00$
and
 $B = -12.42+68.42i 54.30-56.56i -9.93+00.88i 18.32+04.76i -27.30-00.01i -4.40+09.97i 5.31+23.63i 9.43+01.41i .$
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.

10.1Program Text

Program Text (f07hpfe.f90)

10.2Program Data

Program Data (f07hpfe.d)

10.3Program Results

Program Results (f07hpfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017