NAG Library Chapter Introduction

F01 – Matrix Operations, Including Inversion

This chapter describes techniques for inverting a general real matrix A and matrices which are positive definite (have all eigenvalues positive) and are either real and symmetric or complex and Hermitian. It is wasteful and uneconomical not to use the appropriate routine when a matrix is known to have one of these special forms. A general routine must be used when the matrix is not known to be positive definite. In most routines the inverse is computed by solving the linear equations Axi=ei, for i=1,2,…,n, where ei is the ith column of the identity matrix.

Routines are given for calculating the approximate inverse, that is solving the linear equations just once, and also for obtaining the accurate inverse by successive iterative corrections of this first approximation. 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 inverse. In practice the storage requirements for the ‘corrected’ inverse routines are about double those of the ‘approximate’ inverse routines, though the extra computer time is not prohibitive since the same matrix and the same LU decomposition is used in every linear equation solution.

Despite the extra work of the ‘corrected’ inverse routines they are superior to the ‘approximate’ inverse routines. A correction provides a means of estimating the number of accurate figures in the inverse or the number of ‘meaningful’ figures relating to the degree of uncertainty in the coefficients of the matrix.

The residual matrix R=AX-I, where X is a computed inverse of A, conveys useful information. Firstly R is a bound on the relative error in X and secondly R<12  guarantees the convergence of the iterative process in the ‘corrected’ inverse routines.

The decision trees for inversion show which routines in Chapter F04 and Chapter F07 should be used for the inversion of other special types of matrices not treated in the chapter.

For real matrices F08AEF (DGEQRF) and F01QJF return QR and RQ factorizations of A respectively and F08BFF (DGEQP3) returns the QR factorization with column interchanges. The corresponding complex routines are F08ASF (ZGEQRF), F01RJF and F08BTF (ZGEQP3) respectively. Routines are also provided to form the orthogonal matrices and transform by the orthogonal matrices following the use of the above routines. F01QGF and F01RGF form the RQ factorization of an upper trapezoidal matrix for the real and complex cases respectively.

F01BLF uses the QR factorization as described in Section 2.1(ii)(a) and is the only routine that explicitly returns a pseudo-inverse. If m≥n, then the routine will calculate the pseudo-inverse Z of the matrix A. If m<n, then the n by m matrix AT should be used. The routine will calculate the pseudo-inverse Z of AT and the required pseudo-inverse will be ZT. The routine also attempts to calculate the rank, r, of the matrix given a tolerance to decide when elements can be regarded as zero. However, should this routine fail due to an incorrect determination of the rank, the singular value decomposition method (described below) should be used.

F08KBF (DGESVD) and F08KPF (ZGESVD) compute the singular value decomposition as described in Section 2 for real and complex matrices respectively. If A has rank r≤k=minm,n then the k-r smallest singular values will be negligible and the pseudo-inverse of A can be obtained as Z=PΣ-1QT as described in Section 2. If the rank of A is not known in advance it can be estimated from the singular values (see Section 2.4 in the F04 Chapter Introduction). In the real case with m≥n, F02WDF provides details of the QR factorization or the singular value decomposition depending on whether or not A is of full rank and for some problems provides an attractive alternative to F08KBF (DGESVD). For large sparse matrices, leading terms in the singular value decomposition can be computed using routines from Chapter F12.