NAG Library Chapter Introduction
F04 – Simultaneous Linear Equations
1 Scope of the Chapter
This chapter is concerned with the solution of the matrix equation AX=B, where B may be a single vector or a matrix of multiple right-hand sides. The matrix A may be real, complex, symmetric, Hermitian, positive definite, positive definite Toeplitz or banded. It may also be rectangular, in which case a least squares solution is obtained.
For a general introduction to sparse systems of equations, see the
F11 Chapter Introduction, which
currently
provides routines for large sparse systems.
Some routines for sparse problems are also included in this chapter; they are described in
Section 3.4.
2 Background to the Problems
A set of linear equations may be written in the form
where the known matrix
A, with real or complex coefficients, is of size
m by
n (
m rows and
n columns), the known right-hand vector
b has
m components (
m rows and one column), and the required solution vector
x has
n components (
n rows and one column). There may also be
p vectors
bi, for
i=1,2,…,p, on the right-hand side and the equations may then be written as
the required matrix
X having as its
p columns the solutions of
Axi=bi, for
i=1,2,…,p. Some routines deal with the latter case, but for clarity only the case
p=1 is discussed here.
The most common problem, the determination of the unique solution of
Ax=b, occurs when
m=n and
A is not singular, that is
rankA=n. 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
Ax≃b required when
m>n and
rankA=n, i.e., the determination of the vector
x which minimizes the norm of the residual vector
r=b-Ax. All other cases are rank deficient, and they are treated in
Section 2.3.
2.1 Unique Solution of Ax=b
Most routines in this chapter solve this particular problem. The computation starts with the triangular decomposition
A=PLU, 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
the first by forward-substitution and the second by back-substitution.
If A is real symmetric and positive definite, U=LT, while if A is complex Hermitian and positive definite, U=LH; in both these cases P is the identity matrix (i.e., no permutations are necessary). In all other cases either U or L has unit diagonal elements.
Due to rounding errors the computed ‘solution’
x0, 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 x0 may have few or even no figures in common, and at this stage there is no means of estimating the ‘accuracy’ of x0.
There are three possible approaches to estimating the accuracy of a computed solution.
One way to do so, and to ‘correct’
x0 when this is meaningful (see next paragraph), involves computing the residual vector r=b-Ax0 in extended precision arithmetic, and obtaining a correction vector d by solving PLUd=r. The new approximate solution x0+d is usually more accurate and the correcting process is repeated until (a) further corrections are negligible or (b) they show no further decrease.
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 less than full precision.
The first correction vector d will then give some useful information about the number of figures in the ‘solution’ which probably remain unchanged with respect to maximum possible uncertainties in the coefficients.
An alternative
approach to assessing the accuracy of the solution is to compute or estimate the
condition number of
A, defined as
Roughly speaking, errors or uncertainties in
A or
b may be amplified in the solution by a factor
κA. Thus, for example, if the data in
A and
b are only accurate to
5 digits and
κA≈103, then the solution cannot be guaranteed to have more than
2 correct digits. If
κA≥105, the solution may have no meaningful digits.
To be more precise, suppose that
Here
δA and
δb represent perturbations to the matrices
A and
b which cause a perturbation
δx in the solution. We can define measures of the relative sizes of the perturbations in
A,
b and
x as
Then
provided that
κAρA<1. Often
κAρA≪1 and then the bound effectively simplifies to
Hence, if we know
κA,
ρA and
ρb, we can compute a bound on the relative errors in the solution. Note that
ρA,
ρb and
ρx are defined in terms of the norms of
A,
b and
x. If
A,
b or
x contains elements of widely differing magnitude, then
ρA,
ρb and
ρx will be dominated by the errors in the larger elements, and
ρx will give no information about the relative accuracy of smaller elements of
x.
A third
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.
2.2 The Least Squares Solution of Ax≃b, m>n, rankA=n
The least squares solution is the vector
x^ which minimizes the sum of the squares of the residuals,
The solution is obtained in two steps.
| (a) |
Householder transformations are used to reduce A to ‘simpler form’ via the equation QA=R, where R has the appearance
with R^ a nonsingular upper triangular n by n matrix and 0 a zero matrix of shape m-n by n. Similar operations convert b to Qb=c, where
with c1 having n rows and c2 having (m-n) rows. |
| (b) |
The required least squares solution is obtained by back-substitution in the equation
|
Again due to rounding errors the computed
x^0 is only an approximation to the required
x^, but as in
Section 2.1, this can be improved by ‘iterative refinement’. The first correction
d is the solution of the least squares problem
and since the matrix
A is unchanged, this computation takes less time than that of the original
x^0. The process can be repeated until further corrections are
(a) negligible or
(b) show no further decrease.
2.3 Rank-deficient Cases
If, in the least squares problem just discussed, rankA<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 x2, among all those x for which b-Ax2 is a minimum.
This can be computed from the Singular Value Decomposition (SVD) of
A, in which
A is factorized as
where
Q is an
m by
n matrix with orthonormal columns,
P is an
n by
n orthogonal matrix and
D is an
n by
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:
The columns of
Q and
P are called respectively the left and right singular vectors of
A. If the singular values
dr+1,…,dn are zero or negligible, but
dr 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
Ax≃b is given by
where
D† is the diagonal matrix with diagonal elements
d1-1,d2-1,…,dr-1,0,…,0.
The SVD may also be used to find solutions to the homogeneous system of equations
Ax=0, where
A is
m by
n. Such solutions exist if and only if
rankA<n, and are given by
where the
αi are arbitrary numbers and the
pi 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 x^+x, where x^ is the minimal length least squares solution and x is any solution of the homogeneous system of equations Ax=0.
2.4 The Rank of a Matrix
In theory the rank is r if 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:
On a computer with
7 decimal digits of precision the computed singular values were
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
aij=1/i+j-1, the singular values are
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:
where A is m by n and B is p by n, with p≤n≤m+p. The equations Bx=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
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
A
B
has full column rank n. (For linear least squares problems with inequality constraints, refer to Chapter E04.)
- General Gauss–Markov linear model problems:
where A is m by n and B is m by p, with n≤m≤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:
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 routines in this chapter can also be used to calculate the inverse of a square matrix
A by solving the equation
where
I is the identity matrix. However, solving the equations
AX=B by calculation of the inverse of the coefficient matrix
A, i.e., by
X=A-1B, is
definitely not recommended.
Similar remarks apply to the calculation of the pseudo-inverse of a singular or rectangular matrix.
3 Recommendations on Choice and Use of Available Routines
3.1 Black Box and General Purpose Routines
Most of the routines in this chapter are categorised either as Black Box routines or general purpose routines.
Black Box routines solve the equations Axi=bi, for i=1,2,…,p, in a single call with the matrix A and the right-hand sides, bi, being supplied as data. These are the simplest routines to use and are suitable when all the right-hand sides are known in advance and do not occupy too much storage.
General purpose routines, in general, require a previous call to a routine in
Chapters F01,
F03 or
F07 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 routines simply call a factorization routine and then a general purpose routine to solve the equations.
The routine
F04QAF which uses an iterative method for sparse systems of equations does not fit easily into this categorisation, but is classified as a general purpose routine in the decision trees and indexes.
3.2 Systems of Linear Equations
Most of the routines in this chapter solve linear equations
Ax=b when
A is
n by
n and a unique solution is expected (see
Section 2.1). 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).
It must be emphasised that it is a waste of computer time and space to use an inappropriate routine, for example one for the complex case when the equations are real. It is also unsatisfactory to use the special routines for a positive definite matrix if this property is not known in advance.
Routines are given for calculating the
approximate solution, that is solving the linear equations just once, and also for obtaining the
accurate solution by successive iterative corrections of this first approximation using additional precision arithmetic, as described in
Section 2.1. The latter, of course, are more costly in terms of time and storage, since each correction involves the solution of
n sets of linear equations and since the original
A and its
LU decomposition must be stored together with the first and successively corrected approximations to the solution. In practice the storage requirements for the ‘corrected’ routines are about double those of the ‘approximate’ routines, though the extra computer time may not be prohibitive since the same matrix and the same
LU decomposition is used in every linear equation solution.
A number of the Black Box routines in this chapter return estimates of the condition number and the forward error, along with the solution of the equations. But for those routines that do not return a condition estimate two routines are provided –
F04YCF for real matrices,
F04ZCF for complex matrices – which can return a cheap but reliable estimate of
A-1, and hence an estimate of the
condition number
κA (see
Section 2.1). These routines can also be used in conjunction with most of the linear equation solving routines in
Chapter F11: further advice is given in the routine documents.
Other routines 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
The majority of the routines for solving linear least squares problems are to be found in
Chapter F08.
For the case described in
Section 2.2, when
m≥n and a unique least squares solution is expected, there are two routines for a general real
A, one of which (
F04JGF) computes a first approximation and the other (
F04AMF) computes iterative corrections. If it transpires that
rankA<n, so that the least squares solution is not unique, then
F04AMF takes a failure exit, but
F04JGF proceeds to compute the
minimal length solution by using the SVD (see below).
If A is expected to be of less than full rank then one of the routines for calculating the minimal length solution may be used.
For m≫n the use of the SVD is not significantly more expensive than the use of routines based upon the QR factorization.
Problems with linear
equality constraints can be solved by
F08ZAF (DGGLSE) (for real data) or by
F08ZNF (ZGGLSE) (for complex data),
provided that the problems are of full rank. Problems with linear
inequality constraints can be solved by
E04NCF/E04NCA in
Chapter E04.
General Gauss–Markov linear model problems, as formulated in
Section 2.5, can be solved by
F08ZBF (DGGGLM) (for real data) or by
F08ZPF (ZGGGLM) (for complex data).
3.4 Sparse Matrix Routines
Routines specifically for sparse matrices are appropriate only when the number of nonzero elements is very small, less than, say, 10% of the n2 elements of A, and the matrix does not have a relatively small band width.
Chapter F11 contains routines for both the direct and iterative solution of real sparse linear systems. There are two routines in
Chapter F04 for solving sparse linear equations (
F04AXF and
F04QAF).
F04AXF utilizes a factorization of the matrix
A obtained from
F01BRF or
F01BSF, while
F04QAF uses an iterative technique and requires a user-supplied function to compute matrix-vector products
Ac and
ATc for any given vector
c.
F04QAF solves sparse least squares problems by an iterative technique, and also allows the solution of damped (regularised) least squares problems (see the routine document for details).
4 Decision Trees
The name of the routine (if any) that should be used to factorize the matrix A is given in brackets after the name of the routine for solving the equations.
Tree 1: Black Box routines for unique solution of Ax=b (Real matrix)
Tree 2: Black Box routines for unique solution of Ax=b (Complex matrix)
Tree 3: General purpose routines for unique solution of Ax=b (Real matrix)
Tree 4: General purpose routines for unique solution of Ax=b (Complex matrix)
Tree 5: General purpose routines for least squares and homogeneous equations (without constraints)
Note: there are also routines in
Chapter F08 for solving least squares problems..
Note 1: also returns an estimate of the condition number and the forward error.
Note 2: also returns an estimate of the condition number, the forward error and the backward error. Requires additional workspace.
Note 3: for a single right-hand side only.
5 Functionality Index
| Black Box routines, Ax = b, | | |
| complex general band matrix | | F04CBF |
| complex general tridiagonal matrix | | F04CCF |
| complex Hermitian matrix, | | |
| complex Hermitian positive definite band matrix | | F04CFF |
| complex Hermitian positive definite matrix, | | |
| complex Hermitian positive definite tridiagonal matrix | | F04CGF |
| complex symmetric matrix, | | |
| real general band matrix | | F04BBF |
| multiple right-hand sides | | |
| iterative refinement using additional precision | | F04AEF |
| iterative refinement using additional precision | | F04ATF |
| multiple right-hand sides, standard precision | | F04BAF |
| real general tridiagonal matrix | | F04BCF |
| real symmetric positive definite band matrix | | F04BFF |
| real symmetric positive definite matrix | | |
| multiple right-hand sides | | |
| iterative refinement using additional precision | | F04ABF |
| iterative refinement using additional precision | | F04ASF |
| real symmetric positive definite matrix, | | |
| multiple right-hand sides, standard precision | | F04BDF |
| real symmetric positive definite Toeplitz matrix | | |
| general right-hand side | | F04FFF |
| real symmetric positive definite tridiagonal matrix | | F04BGF |
| General Purpose routines, Ax = b, | | |
| real almost block-diagonal matrix | | F04LHF |
| real band symmetric positive definite matrix, variable bandwidth | | F04MCF |
| real symmetric positive definite Toeplitz matrix, | | |
| general right-hand side, update solution | | F04MFF |
| Yule–Walker equations, update solution | | F04MEF |
| real tridiagonal matrix | | F04LEF |
| Least squares and Homogeneous Equations, | | |
| m ≥ n, rank = n or minimal solution | | F04JGF |
| rank = n, iterative refinement | | F04AMF |
| norm and condition number estimation | | F04ZCF |
| covariance matrix for linear least squares problems | | F04YAF |
| norm and condition number estimation | | F04YCF |
6 Auxiliary Routines Associated with Library Routine Parameters
None.
7 Routines Withdrawn or Scheduled for Withdrawal
8 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