NAG C Library Chapter Introduction

f04 — Simultaneous Linear Equations

. This is discussed in Section 2.1 below. The next most common problem, discussed in Section 2.2 below, is the determination of the least-squares solution of

A x ≃ b

required when

m > n

and

rank (A) = n

, i.e., the determination of the vector

x

which minimizes the norm of the residual vector

r = b - A x

. All other cases are rank deficient, and they are treated in Section 2.3.

Most of the functions of the chapter are based on those published in the book edited by Wilkinson and Reinsch (1971). We are very grateful to the late Dr J H Wilkinson FRS for his help and interest during the implementation of this chapter of the Library.

2.1 Unique Solution of $A x = b$

Most functions in this chapter solve this particular problem. The computation starts with the triangular decomposition

A = P L U

, where

L

and

U

are respectively lower and upper triangular matrices and

P

is a permutation matrix, chosen so as to ensure that the decomposition is numerically stable. The solution is then obtained by solving in succession the simpler equations

\begin{array}{lcl} L y & = & P^{T} b \\ U x & = & y \end{array}

the first by forward-substitution and the second by back-substitution.

A

is real symmetric and positive-definite,

U = L^{T}

, while if

A

is complex Hermitian and positive-definite,

U = L^{H}

; in both these cases

P

is the identity matrix (i.e., no permutations are necessary). In all other cases either

U

L

has unit diagonal elements.

Due to rounding errors the computed ‘solution’

x_{0}

, say, is only an approximation to the true solution

x

. This approximation will sometimes be satisfactory, agreeing with

x

to several figures, but if the problem is ill-conditioned then

x

and

x_{0}

may have few or even no figures in common, and at this stage there is no means of estimating the ‘accuracy’ of

x_{0}

It must be emphasised that the ‘true’ solution

x

may not be meaningful, that is correct to all figures quoted, if the elements of

A

and

b

are known with certainty only to say

p

figures, where

p

is smaller than the word-length of the computer.

One approach to assessing the accuracy of the solution is to compute or estimate the condition number of

A

, defined as

κ (A) = ‖A‖ ‖A^{- 1}‖ .

Roughly speaking, errors or uncertainties in

A

b

may be amplified in the solution by a factor

κ (A)

. Thus, for example, if the data in

A

and

b

are only accurate to 5 digits and

κ (A) \approx 10^{3}

, then the solution cannot be guaranteed to have more than 2 correct digits. If

κ (A) \geq 10^{5}

, the solution may have no meaningful digits.

To be more precise, suppose that

A x = b and (A + δ A) (x + δ x) = b + δ b .

Here

δ A

and

δ b

represent perturbations to the matrices

A

and

b

which cause a perturbation

δ x

in the solution. We can define measures of the relative sizes of the perturbations in

A

b

and

x

ρ_{A} = \frac{‖δ A‖}{‖A‖}, ρ_{b} = \frac{‖δ b‖}{‖b‖} and ρ_{x} = \frac{‖δ x‖}{‖x‖} respectively.

Then

ρ_{x} \leq \frac{κ (A)}{1 - κ (A) ρ_{A}} (ρ_{A} + ρ_{b})

provided that

κ (A) ρ_{A} < 1

. Often

κ (A) ρ_{A} ≪ 1

and then the bound effectively simplifies to

ρ_{x} \leq κ (A) (ρ_{A} + ρ_{b}) .

Hence, if we know

κ (A)

ρ_{A}

and

ρ_{b}

, we can compute a bound on the relative errors in the solution. Note that

ρ_{A}

ρ_{b}

and

ρ_{x}

are defined in terms of the norms of

A

b

and

x

. If

A

b

x

contains elements of widely differing magnitude, then

ρ_{A}

ρ_{b}

and

ρ_{x}

will be dominated by the errors in the larger elements, and

ρ_{x}

will give no information about the relative accuracy of smaller elements of

x

Another way to obtain useful information about the accuracy of a solution is to solve two sets of equations, one with the given coefficients, which are assumed to be known with certainty to

p

figures, and one with the coefficients rounded to (

p - 1

) figures, and to count the number of figures to which the two solutions agree. In ill-conditioned problems this can be surprisingly small and even zero.

Chapter f07 contains functions for estimating condition numbers and for returning error bounds.

2.2 The Least-squares Solution of $A x ≃ b$ , $m > n$ , $rank A = n$

The least-squares solution is the vector

\hat{x}

which minimizes the sum of the squares of the residuals,

S = {(b - A \hat{x})}^{T} (b - A \hat{x}) = {‖b - A \hat{x}‖}_{2}^{2} .

The solution is obtained in two steps.

Householder Transformations are used to reduce $A$ to ‘simpler form’ via the equation $Q A = R$ , where $R$ has the appearance
$(\frac{\hat{R}}{0})$
with $\hat{R}$ a non-singular upper triangular $n$ by $n$ matrix and 0 a zero matrix of shape $(m - n)$ by $n$ . Similar operations convert $b$ to $Q b = c$ , where
$c = (\frac{c_{1}}{c_{2}})$
with $c_{1}$ having $n$ rows and $c_{2}$ having ( $m - n$ ) rows.
The required least-squares solution is obtained by back-substitution in the equation
$\hat{R} \hat{x} = c_{1} .$

Again due to rounding errors the computed

{\hat{x}}_{0}

is only an approximation to the required

\hat{x}

2.3 Rank-deficient Cases

If, in the least-squares problem just discussed,

rank (A) < n

, then a least-squares solution exists but it is not unique. In this situation it is usual to ask for the least-squares solution ‘of minimal length’, i.e., the vector

x

which minimizes

{‖x‖}_{2}

, among all those

x

for which

{‖b - A x‖}_{2}

is a minimum.

This can be computed from the Singular Value Decomposition (SVD) of

A

, in which

A

is factorized as

A = Q D P^{T}

where

Q

is an

m

n

matrix with orthonormal columns,

P

is an

n

n

orthogonal matrix and

D

is an

n

n

diagonal matrix. The diagonal elements of

D

are called the ‘singular values’ of

A

; they are non-negative and can be arranged in decreasing order of magnitude:

d_{1} \geq d_{2} \geq \dots \geq d_{n} \geq 0 .

The columns of

Q

and

P

are called respectively the left and right singular vectors of

A

. If the singular values

d_{r + 1}, \dots, d_{n}

are zero or negligible, but

d_{r}

is not negligible, then the rank of

A

is taken to be

r

(see also Section 2.4) and the minimal length least-squares solution of

A x ≃ b

is given by

\hat{x} = D^{†} Q^{T} b

where

D^{†}

is the diagonal matrix with diagonal elements

d_{1}^{- 1}, d_{2}^{- 1}, \dots, d_{r}^{- 1}, 0, \dots, 0

The SVD may also be used to find solutions to the homogeneous system of equations

A x = 0

, where

A

m

n

. Such solutions exist if and only if

rank (A) < n

, and are given by

x = \sum_{i = r + 1}^{n} α_{i} p_{i}

where the

α_{i}

are arbitrary numbers and the

p_{i}

are the columns of

P

which correspond to negligible elements of

D

The general solution to the rank-deficient least-squares problem is given by

\hat{x} + x

, where

\hat{x}

is the minimal length least-squares solution and

x

is any solution of the homogeneous system of equations

A x = 0

2.4 The Rank of a Matrix

In theory the rank is

r

n - r

elements of the diagonal matrix

D

of the singular value decomposition are exactly zero. In practice, due to rounding and/or experimental errors, some of these elements have very small values which usually can and should be treated as zero.

For example, the following 5 by 8 matrix has rank 3 in exact arithmetic:

(\begin{array}{r} 22 & 14 & - 1 & - 3 & 9 & 9 & 2 & 4 \\ 10 & 7 & 13 & - 2 & 8 & 1 & - 6 & 5 \\ 2 & 10 & - 1 & 13 & 1 & - 7 & 6 & 0 \\ 3 & 0 & - 11 & - 2 & - 2 & 5 & 5 & - 2 \\ 7 & 8 & 3 & 4 & 4 & - 1 & 1 & 2 \end{array}) .

On a computer with 7 decimal digits of precision the computed singular values were

3.5 \times 10^{1}, 2.0 \times 10^{1}, 2.0 \times 10^{1}, 1.3 \times 10^{- 6}, 5.5 \times 10^{- 7}

and the rank would be correctly taken to be 3.

It is not, however, always certain that small computed singular values are really zero. With the 7 by 7 Hilbert matrix, for example, where

a_{i j} = 1 / (i + j - 1)

, the singular values are

1.7, 2.7 \times 10^{- 1}, 2.1 \times 10^{- 2}, 1.0 \times 10^{- 3}, 2.9 \times 10^{- 5}, 4.9 \times 10^{- 7}, 3.5 \times 10^{- 9} .

Here there is no clear cut-off between small (i.e., negligible) singular values and larger ones. In fact, in exact arithmetic, the matrix is known to have full rank and none of its singular values is zero. On a computer with 7 decimal digits of precision, the matrix is effectively singular, but should its rank be taken to be 6, or 5, or 4?

It is therefore impossible to give an infallible rule, but generally the rank can be taken to be the number of singular values which are neither zero nor very small compared with other singular values. For example, if there is a sharp decrease in singular values from numbers of order unity to numbers of order

10^{- 7}

, then the latter will almost certainly be zero in a machine in which 7 significant decimal figures is the maximum accuracy. Similarly for a least-squares problem in which the data is known to about four significant figures and the largest singular value is of order unity then a singular value of order

10^{- 4}

or less should almost certainly be regarded as zero.

It should be emphasised that rank determination and least-squares solutions can be sensitive to the scaling of the matrix. If at all possible the units of measurement should be chosen so that the elements of the matrix have data errors of approximately equal magnitude.

2.5 Generalized Linear Least-squares Problems

The simple type of linear least-squares problem described in Section 2.2 can be generalized in various ways.

Linear least-squares problems with equality constraints:
$find x to minimize S = {‖c - A x‖}_{2}^{2} subject to B x = d,$
where $A$ is $m$ by $n$ and $B$ is $p$ by $n$ , with $p \leq n \leq m + p$ . The equations $B x = d$ may be regarded as a set of equality constraints on the problem of minimizing $S$ . Alternatively the problem may be regarded as solving an overdetermined system of equations
$(\begin{matrix} A \\ B \end{matrix}) x = (\begin{matrix} c \\ d \end{matrix}),$
where some of the equations (those involving $B$ ) are to be solved exactly, and the others (those involving $A$ ) are to be solved in a least-squares sense. The problem has a unique solution on the assumptions that $B$ has full row rank $p$ and the matrix $(\begin{matrix} A \\ B \end{matrix})$ has full column rank $n$ . (For linear least-squares problems with inequality constraints, refer to Chapter e04.)
General Gauss–Markov linear model problems:
$minimize {‖y‖}_{2} subject to d = A x + B y,$
where $A$ is $m$ by $n$ and $B$ is $m$ by $p$ , with $n \leq m \leq n + p$ . When $B = I$ , the problem reduces to an ordinary linear least-squares problem. When $B$ is square and nonsingular, it is equivalent to a weighted linear least-squares problem:
$find x to minimize {‖B^{- 1} (d - A x)‖}_{2} .$
The problem has a unique solution on the assumptions that $A$ has full column rank $n$ , and the matrix $(A, B)$ has full row rank $m$ .

2.6 Calculating the Inverse of a Matrix

The functions in this chapter can also be used to calculate the inverse of a square matrix

A

by solving the equation

A X = I

where

I

is the identity matrix. However, solving the equations

A X = B

by calculation of the inverse of the coefficient matrix

A

, i.e., by

X = A^{- 1} B

, is definitely not recommended.

3 Recommendations on Choice and Use of Available Functions

3.1 Black Box and General Purpose Functions

Most of the functions in this chapter are categorised as Black Box functions or general purpose functions.

Black Box functions solve the equations

A x_{i} = b_{i}

, for

i = 1, 2, \dots, p

, in a single call with the matrix

A

and the right-hand sides

b_{i}

being supplied as data. These are the simplest functions to use and are suitable when all the right-hand sides are known in advance and do not occupy too much storage.

General purpose functions, in general, require a previous call to a function in Chapters f01 or f03 to factorize the matrix

A

. This factorization can then be used repeatedly to solve the equations for one or more right-hand sides which may be generated in the course of the computation. The Black Box functions simply call a factorization function and then a general purpose function to solve the equations.

3.2 Systems of Linear Equations

Most of the functions in this chapter solve linear equations

A x = b

when

A

n

n

and a unique solution is expected (case

2.1

). If this turns out to be untrue the functions go to a failure exit. The matrix

A

may be ‘general’ real or complex, or may have special structure or properties, e.g., it may be banded, tridiagonal, almost block-diagonal, sparse, symmetric, Hermitian, positive-definite (or various combinations of these). For some of the combinations see Chapter f07. nag_real_cholesky_skyline_solve (f04mcc) (which needs to be preceded by a call to nag_real_cholesky_skyline (f01mcc)) can be used for the solution of variable band-width (skyline) positive-definite systems.

It must be emphasised that it is a waste of computer time and space to use an inappropriate function, for example one for the complex case when the equations are real. It is also unsatisfactory to use the special functions for a positive-definite matrix if this property is not known in advance.

Other functions for solving linear equation systems, computing inverse matrices, and estimating condition numbers can be found in Chapter f07, which contains LAPACK software.

3.3 Linear Least-squares Problems

Functions for solving linear least-squares problems using the

Q R

factorization or the SVD can be found in
Chapters f01, f02 and f08. When

m \geq n

and a unique solution is expected, the

Q R

factorization can be used, otherwise the

Q R

factorization with pivoting, or the SVD should be used. For

m ≫ n

, the SVD is not significantly more expensive than the

Q R

factorization. See Chapter f08 for further discussion.

Problems with linear equality constraints can be solved by functions in Chapter f08 provided that the problems are of full rank. Problems with linear inequality constraints can be solved by nag_opt_lin_lsq (e04ncc) in Chapter e04.

General Gauss–Markov linear model problems, as formulated in Section 2.5, can be solved by functions in Chapter f08.

3.4 Sparse Matrix Functions

For the solution of sparse linear equations see Chapter f11.

4 Decision Trees

If at any stage the answer to a question is ‘Don't know’ this should be read as ‘No’.

The name of the function (if any) that should be used to factorize the matrix

A

is given in brackets after the name of the function for solving the equations.

Tree 1: Black Box functions for unique solution of $A x = b$

Is $A$ a real matrix?	_ yes	f04arc
no \|
f04adc

Tree 2: General purpose functions for unique solution of $A x = b$

Is $A$ a real matrix?	_ yes	Is $A$ a symmetric positive-definite matrix?	_ yes	Is $A$ a variable band matrix?	_ yes	f04mcc (f01mcc)
\|		\|		no \|
\|		\|		f04agc (f03aec)
\|		no \|
\|		f04ajc (f03afc)
no \|
Is $A$ a Hermitian positive-definite matrix?	_ yes	f04awc (f01bnc)
no \|
f04akc (f03ahc)

Tree 3: General purpose functions for unique solution of $A x = b$ (Complex matrix)

Is $A$ Hermitian?	_ yes	Is $A$ positive-definite?	_ yes	Is $A$ a band matrix?	_ yes	f07hsc (f07hrc)
\|		\|		no \|
\|		\|		Is one triangle of $A$ stored as a linear array?	_ yes	f07gsc (f07grc)
\|		\|		no \|
\|		\|		f07fsc (f07frc)
\|		no \|
\|		Is one triangle of $A$ stored as a linear array?	_ yes	f07psc (f07prc)
\|		no \|
\|		f07msc (f07mrc)
no \|
Is $A$ symmetric?	_ yes	Is one triangle of $A$ stored as a linear array?	_ yes	f07qsc (f07qrc)
\|		no \|
\|		f07nsc) (f07nrc)
no \|
Is $A$ triangular?	_ yes	Is $A$ a band matrix?	_ yes	f07vsc
\|		no \|
\|		Is $A$ stored as a linear array?	_ yes	f07usc
\|		no \|
\|		f07tsc
no \|
Is $A$ a band matrix?	_ yes	f07bsc (f07brc)
no \|
f07asc (f07arc)

5 Index

Black Box functions, $A x = b$ ,
Complex matrix	f04adc
Real matrix,
Single right-hand side	f04arc
General Purpose functions, $A x = b$ ,
Complex Hermitian positive-definite matrix	f04awc
Complex matrix	f04akc
Real band symmetric positive-definite matrix,
variable bandwidth	f04mcc
Real matrix	f04ajc
Real symmetric positive-definite matrix	f04agc

6 Functions Withdrawn or Scheduled for Withdrawal

None.

7 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Lawson C L and Hanson R J (1974) Solving Least-squares Problems Prentice–Hall

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

f04 Chapter Introduction (pdf version)

Chapter Contents

NAG C Library Contents

NAG C Library Chapter Introductionf04 — Simultaneous Linear Equations

Contents

NAG C Library Chapter Introduction

f04 — Simultaneous Linear Equations