F01 Chapter Introduction : NAG Library, Mark 26

This chapter provides facilities for four types of problem:

(i)	Matrix Inversion
(ii)	Matrix Factorizations
(iii)	Matrix Arithmetic and Manipulation
(iv)	Matrix Functions

See Sections 2.1, 2.2, 2.3 and 2.4 where these problems are discussed.

The routines in this section perform matrix factorizations which are required for the solution of systems of linear equations with various special structures. A few routines which perform associated computations are also included.

Other routines for matrix factorizations are to be found in Chapters F07, F08 and F11.

This section also contains a few routines associated with eigenvalue problems (see Chapter F02). (Historical note: this section used to contain many more such routines, but they have now been superseded by routines in Chapter F08.)

The intention of routines in this section (sub-chapters F01C, F01V and F01Z) is to cater for some of the commonly occurring operations in matrix manipulation, i.e., transposing a matrix or adding part of one matrix to another, and for conversion between different storage formats,such as conversion between rectangular band matrix storage and packed band matrix storage. For vector or matrix-vector or matrix-matrix operations refer to Chapters F06 and F16.

Given a square matrix

A

, the matrix function

f (A)

is a matrix with the same dimensions as

A

which provides a generalization of the scalar function

f

.

If

A

has a full set of eigenvectors

V

then

A

can be factorized as

A = V D V^{- 1},

where

D

is the diagonal matrix whose diagonal elements,

d_{i}

, are the eigenvalues of

A

.

f (A)

is given by

f (A) = V f (D) V^{- 1},

where

f (D)

is the diagonal matrix whose

i

th diagonal element is

f (d_{i})

.

In general,

A

may not have a full set of eigenvectors. The matrix function can then be defined via a Cauchy integral. For

A \in ℂ^{n \times n}

,

f (A) = \frac{1}{2 π i} \int_{Γ} f (z) {(z I - A)}^{- 1} d z,

where

Γ

is a closed contour surrounding the eigenvalues of

A

, and

f

is analytic within

Γ

.

Some matrix functions are defined implicitly. A matrix logarithm is a solution

X

to the equation

e^{X} = A .

In general

X

is not unique, but if

A

has no eigenvalues on the closed negative real line then a unique principal logarithm exists whose eigenvalues have imaginary part between

π

and

- π

. Similarly, a matrix square root is a solution

X

to the equation

X^{2} = A .

If

A

has no eigenvalues on the closed negative real line then a unique principal square root exists with eigenvalues in the right half-plane. If

A

has a vanishing eigenvalue then

\log (A)

cannot be computed. If the vanishing eigenvalue is defective (its algebraic multiplicity exceeds its geometric multiplicity, or equivalently it occurs in a Jordan block of size greater than

1

) then the square root cannot be computed. If the vanishing eigenvalue is semisimple (its algebraic and geometric multiplicities are equal, or equivalently it occurs only in Jordan blocks of size

1

) then a square root can be computed.

Algorithms for computing matrix functions are usually tailored to a specific function. Currently Chapter F01 contains routines for calculating the exponential, logarithm, sine, cosine, sinh, cosh, square root and general real power of both real and complex matrices. In addition there are routines to compute a general function of real symmetric and complex Hermitian matrices and a general function of general real and complex matrices.

The Fréchet derivative of a matrix function

f (A)

in the direction of the matrix

E

is the linear function mapping

E

to

L_{f} (A, E)

such that

f (A + E) - f (A) - L_{f} (A, E) = O (‖E‖) .

The Fréchet derivative measures the first-order effect on

f (A)

of perturbations in

A

. Chapter F01 contains routines for calculating the Fréchet derivative of the exponential, logarithm and real powers of both real and complex matrices.

The condition number of a matrix function is a measure of its sensitivity to perturbations in the data. The absolute condition number measures these perturbations in an absolute sense, and is defined by

{cond}_{abs} (f, A) ≔ \lim_{ε \to 0} \sup_{\{‖E‖ \to 0\}} \frac{‖f (A + E) - f (A)‖}{ε} .

The relative condition number, which is usually of more interest, measures these perturbations in a relative sense, and is defined by

{cond}_{rel} (f, A) = {cond}_{abs} (f, A) \frac{‖A‖}{‖f (A)‖} .

The absolute and relative condition numbers can be expressed in terms of the norm of the Fréchet derivative by

{cond}_{abs} (f, A) = \max_{E \neq 0} \frac{‖L (A, E)‖}{‖E‖},

{cond}_{rel} (f, A) = \frac{‖A‖}{‖f (A)‖} \max_{E \neq 0} \frac{‖L (A, E)‖}{‖E‖} .

Chapter F01 contains routines for calculating the condition number of the matrix exponential, logarithm, sine, cosine, sinh, cosh, square root and general real power of both real and complex matrices. It also contains routines for estimating the condition number of a general function of a real or complex matrix.

Note: before using any routine for matrix inversion, consider carefully whether it is really needed.

Although the solution of a set of linear equations

A x = b

can be written as

x = A^{- 1} b

, the solution should never be computed by first inverting

A

and then computing

A^{- 1} b

; the routines in Chapters F04 or F07 should always be used to solve such sets of equations directly; they are faster in execution, and numerically more stable and accurate. Similar remarks apply to the solution of least squares problems which again should be solved by using the routines in Chapters F04 and F08 rather than by computing a pseudo-inverse.

(a)

Nonsingular square matrices of order

n

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

A x_{i} = e_{i}

, for

i = 1, 2, \dots, n

, where

e_{i}

is the

i

th 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

L U

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

L U

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 = A X - 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‖ < \frac{1}{2}

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.

(b)

General real rectangular matrices

For real matrices f08aef (dgeqrf) and f01qjf return

Q R

and

R Q

factorizations of

A

respectively and f08bff (dgeqp3) returns the

Q R

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

R Q

factorization of an upper trapezoidal matrix for the real and complex cases respectively.

f01blf uses the

Q R

factorization as described in Section 2.1(ii)(a) and is the only routine that explicitly returns a pseudo-inverse. If

m \geq n

, then the routine will calculate the pseudo-inverse

A^{+}

of the matrix

A

. If

m < n

, then the

n

by

m

matrix

A^{T}

should be used. The routine will calculate the pseudo-inverse

Z = {(A^{T})}^{+} = {(A^{+})}^{T}

of

A^{T}

and the required pseudo-inverse will be

Z^{T}

. 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 \leq k = \min (m, n)

then the

k - r

smallest singular values will be negligible and the pseudo-inverse of

A

can be obtained as

A^{+} = V Σ^{- 1} U^{T}

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 \geq n

, f08aef (dgeqrf) followed by f02wuf provide details of the

Q R

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.

Each of these routines serves a special purpose required for the solution of sets of simultaneous linear equations or the eigenvalue problem. For further details you should consult Sections 3 or 4 in the F02 Chapter Introduction or Sections 3 or 4 in the F04 Chapter Introduction.

f01brf and f01bsf are provided for factorizing general real sparse matrices. A more recent algorithm for the same problem is available through f11mef. For factorizing real symmetric positive definite sparse matrices, see f11jaf. These routines should be used only when

A

is not banded and when the total number of nonzero elements is less than 10% of the total number of elements. In all other cases either the band routines or the general routines should be used.

The routines in the F01C section are designed for the general handling of

m

by

n

matrices. Emphasis has been placed on flexibility in the argument specifications and on avoiding, where possible, the use of internally declared arrays. They are therefore suited for use with large matrices of variable row and column dimensions. Routines are included for the addition and subtraction of sub-matrices of larger matrices, as well as the standard manipulations of full matrices. Those routines involving matrix multiplication may use additional-precision arithmetic for the accumulation of inner products. See also Chapter F06.

The routines in the F01V (LAPACK) and F01Z section are designed to allow conversion between full storage format and one of the packed storage schemes required by some of the routines in Chapters F02, F04, F06, F07 and F08.

Routines with NAG name beginning F01V may be called either by their NAG names or by their LAPACK names. When using the NAG Library, the double precision form of the LAPACK name must be used (beginning with D- or Z-).

References to Chapter F01 routines in the manual normally include the LAPACK double precision names, for example, f01vef (dtrttf).

The LAPACK routine names follow a simple scheme (which is similar to that used for the BLAS in Chapter F06). Most names have the structure XYYTZZ, where the components have the following meanings:

– the initial letter, X, indicates the data type (real or complex) and precision:

S – real, single precision (in Fortran, 4 byte length REAL)
D – real, double precision (in Fortran, 8 byte length REAL)
C – complex, single precision (in Fortran, 8 byte length COMPLEX)
Z – complex, double precision (in Fortran, 16 byte length COMPLEX)

– the fourth letter, T, indicates that the routine is performing a storage scheme transformation (conversion)

– the letters YY indicate the original storage scheme used to store a triangular part of the matrix

A

, while the letters ZZ indicate the target storage scheme of the conversion (YY cannot equal ZZ since this would do nothing):

TF – Rectangular Full Packed Format (RFP)
TP – Packed Format
TR – Full Format

f01ecf and f01fcf compute the matrix exponential,

e^{A}

, of a real and complex square matrix

A

respectively. If estimates of the condition number of the matrix exponential are required then f01jgf and f01kgf should be used. If Fréchet derivatives are required then f01jhf and f01khf should be used.

f01edf and f01fdf compute the matrix exponential,

e^{A}

, of a real symmetric and complex Hermitian matrix respectively. If the matrix is real symmetric, or complex Hermitian then it is recommended that f01edf, or f01fdf be used as they are more efficient and, in general, more accurate than f01ecf and f01fcf.

f01ejf and f01fjf compute the principal matrix logarithm,

\log (A)

, of a real and complex square matrix

A

respectively. If estimates of the condition number of the matrix logarithm are required then f01jjf and f01kjf should be used. If Fréchet derivatives are required then f01jkf and f01kkf should be used.

f01ekf and f01fkf compute the matrix exponential, sine, cosine, sinh or cosh of a real and complex square matrix

A

respectively. If the matrix exponential is required then it is recommended that f01ecf or f01fcf be used as they are, in general, more accurate than f01ekf and f01fkf. If estimates of the condition number of the matrix function are required then f01jaf and f01kaf should be used.

f01elf and f01emf compute the matrix function,

f (A)

, of a real square matrix. f01flf and f01fmf compute the matrix function of a complex square matrix. The derivatives of

f

are required for these computations. f01elf and f01flf use numerical differentiation to obtain the derivatives of

f

. f01emf and f01fmf use derivatives you have supplied. If estimates of the condition number are required but you are not supplying derivatives then f01jbf and f01kbf should be used. If estimates of the condition number of the matrix function are required and you are supplying derivatives of

f

, then f01jcf and f01kcf should be used.

If the matrix

A

is real symmetric or complex Hermitian then it is recommended that to compute the matrix function,

f (A)

, f01eff and f01fff are used respectively as they are more efficient and, in general, more accurate than f01elf, f01emf, f01flf and f01fmf.

f01gaf and f01haf compute the matrix function

e^{t A} B

for explicitly stored dense real and complex matrices

A

and

B

respectively while f01gbf and f01hbf compute the same using reverse communication. In the latter case, control is returned to you. You should calculate any required matrix-matrix products and then call the routine again. See Section 3.3.3 in How to Use the NAG Library and its Documentation for further information.

f01enf and f01fnf compute the principal square root

A^{1 / 2}

of a real and complex square matrix

A

respectively. If

A

is complex and upper triangular then f01fpf should be used. If

A

is real and upper quasi-triangular then f01epf should be used. If estimates of the condition number of the matrix square root are required then f01jdf and f01kdf should be used.

f01eqf and f01fqf compute the matrix power

A^{p}

, where

p \in ℝ

, of real and complex matrices respectively. If estimates of the condition number of the matrix power are required then f01jef and f01kef should be used. If Fréchet derivatives are required then f01jff and f01kff should be used.

The decision trees show the routines in this chapter and in Chapter F04, Chapter F07 and Chapter F08 that should be used for inverting matrices of various types. They also show which routine should be used to calculate various matrix functions.

(i) Matrix Inversion:

Note 1: the inverse of a band matrix

A

does not in general have the same shape as

A

, and no routines are provided specifically for finding such an inverse. The matrix must either be treated as a full matrix, or the equations

A X = B

must be solved, where

B

has been initialized to the identity matrix

I

. In the latter case, see the decision trees in Section 4 in the F04 Chapter Introduction.

Note 2: by ‘guaranteed accuracy’ we mean that the accuracy of the inverse is improved by use of the iterative refinement technique using additional precision.

(ii) Matrix Factorizations: see the decision trees in Section 4 in the F02 and F04 Chapter Introductions.

(iii) Matrix Arithmetic and Manipulation: not appropriate.

(iv) Matrix Functions:

None.

The following lists all those routines that have been withdrawn since Mark 19 of the Library or are scheduled for withdrawal at one of the next two marks.

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

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford

Wilkinson J H (1977) Some recent advances in numerical linear algebra The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press

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

Is $A$ an $n$ by $n$ matrix of rank $n$ ?			Is $A$ a real matrix?			see Tree 2
		yes			yes
	no			no
			see Tree 3


see Tree 4

Is $A$ a band matrix?			See Note 1.
Is $A$ a band matrix?		yes	See Note 1.
	no
Is $A$ symmetric?			Is $A$ positive definite?			Do you want guaranteed accuracy? (See Note 2)			f01abf
Is $A$ symmetric?		yes	Is $A$ positive definite?		yes	Do you want guaranteed accuracy? (See Note 2)		yes	f01abf
	no			no			no
						Is one triangle of $A$ stored as a linear array?			f07gdf and f07gjf
						Is one triangle of $A$ stored as a linear array?		yes	f07gdf and f07gjf
							no
						f01adf or f07fdf and f07fjf
						f01adf or f07fdf and f07fjf

			Is one triangle of $A$ stored as a linear array?			f07pdf and f07pjf
			Is one triangle of $A$ stored as a linear array?		yes	f07pdf and f07pjf
				no
			f07mdf and f07mjf
			f07mdf and f07mjf

Is $A$ triangular?			Is $A$ stored as a linear array?			f07ujf
Is $A$ triangular?		yes	Is $A$ stored as a linear array?		yes	f07ujf
	no			no
			f07tjf
			f07tjf

Do you want guaranteed accuracy? (See Note 2)			f07abf
Do you want guaranteed accuracy? (See Note 2)		yes	f07abf
	no
f07adf and f07ajf
f07adf and f07ajf

Is $A$ a band matrix?			See Note 1.
		yes
	no
Is $A$ Hermitian?			Is $A$ positive definite?			Is one triangle of $A$ stored as a linear array?			f07grf and f07gwf
		yes			yes			yes
	no			no			no
						f07frf and f07fwf


			Is one triangle $A$ stored as a linear array?			f07prf and f07pwf
					yes
				no
			f07mrf and f07mwf


Is $A$ symmetric?			Is one triangle of $A$ stored as a linear array?			f07qrf and f07qwf
		yes			yes
	no			no
			f07nrf and f07nwf


Is $A$ triangular?			Is $A$ stored as a linear array?			f07uwf
		yes			yes
	no			no
			f07twf


f07anf or f07arf and f07awf

Is $A$ a complex matrix?			Is $A$ of full rank?			Is $A$ an $m$ by $n$ matrix with $m < n$ ?			f01rjf and f01rkf
		yes			yes			yes
	no			no			no
						f08asf and f08auf or f08atf


			f08kpf


Is $A$ of full rank?			Is $A$ an $m$ by $n$ matrix with $m < n$ ?			f01qjf and f01qkf
		yes			yes
	no			no
			f08aef and f08agf or f08aff


Is $A$ an $m$ by $n$ matrix with $m < n$ ?			f08kbf
		yes
	no
Is reliability more important than efficiency?			f08kbf
		yes
	no
f01blf

Withdrawn Routine	Mark of Withdrawal	Replacement Routine(s)
f01maf	19	f11jaf

NAG Library Chapter Introduction

F01 (matop)
Matrix Operations, Including Inversion

▸▿ Contents

1

Scope of the Chapter

2

Background to the Problems

2.1

Matrix Inversion

2.2

Matrix Factorizations

2.3

Matrix Arithmetic and Manipulation

2.4

Matrix Functions

3

Recommendations on Choice and Use of Available Routines

3.1

Matrix Inversion

3.2

Matrix Factorizations

3.3

Matrix Arithmetic and Manipulation

3.3.1

NAG Names and LAPACK Names

3.4

Matrix Functions

4

Decision Trees

Tree 1

Tree 2: Inverse of a real n by n matrix of full rank

Tree 3: Inverse of a complex n by n matrix of full rank

Tree 4: Pseudo-inverses

Tree 5: Matrix functions $f (A)$ of an n by n real matrix $A$

Tree 6: Matrix functions $f (A)$ of an n by n complex matrix $A$

5

Functionality Index

6

Auxiliary Routines Associated with Library Routine Arguments

7

Routines Withdrawn or Scheduled for Withdrawal

8

References

Is $e^{t A} B$ required?			Is $A$ stored in dense format?			f01gaf
		yes			yes
	no			no
			f01gbf


Is $A$ real symmetric?			Is $e^{A}$ required?			f01edf
		yes			yes
	no			no
			f01eff


Is $\cos (A)$ or $\cosh (A)$ or $\sin (A)$ or $\sinh (A)$ required?			Is the condition number of the matrix function required?			f01jaf
		yes			yes
	no			no
			f01ekf


Is $\log (A)$ required?			Is the condition number of the matrix logarithm required?			f01jjf
		yes			yes
	no			no
			Is the Fréchet derivative of the matrix logarithm required?			f01jkf
					yes
				no
			f01ejf


Is $\exp (A)$ required?			Is the condition number of the matrix exponential required?			f01jgf
		yes			yes
	no			no
			Is the Fréchet derivative of the matrix exponential required?			f01jhf
					yes
				no
			f01ecf


Is $A^{1 / 2}$ required?			Is the condition number of the matrix square root required?			f01jdf
		yes			yes
	no			no
			Is the matrix upper quasi-triangular?			f01epf
					yes
				no
			f01enf


Is $A^{p}$ required?			Is the condition number of the matrix power required?			f01jef
		yes			yes
	no			no
			Is the Fréchet derivative of the matrix power required?			f01jff
					yes
				no
			f01eqf


$f (A)$ will be computed. Will derivatives of $f$ be supplied by the user?			Is the condition number of the matrix function required?			f01jcf
		yes			yes
	no			no
			f01emf


Is the condition number of the matrix function required?			f01jbf
		yes
	no
f01elf

Is $e^{t A} B$ required?			Is $A$ stored in dense format?			f01haf
		yes			yes
	no			no
			f01hbf


Is $A$ complex Hermitian?			Is $e^{A}$ required?			f01fdf
		yes			yes
	no			no
			f01fff


Is $\cos (A)$ or $\cosh (A)$ or $\sin (A)$ or $\sinh (A)$ required?			Is the condition number of the matrix function required?			f01kaf
		yes			yes
	no			no
			f01fkf


Is $\log (A)$ required?			Is the condition number of the matrix logarithm required?			f01kjf
		yes			yes
	no			no
			Is the Fréchet derivative of the matrix logarithm required?			f01kkf
					yes
				no
			f01fjf


Is $\exp (A)$ required?			Is the condition number of the matrix exponential required?			f01kgf
		yes			yes
	no			no
			Is the Fréchet derivative of the matrix exponential required?			f01khf
					yes
				no
			f01fcf


Is $A^{1 / 2}$ required?			Is the condition number of the matrix square root required?			f01kdf
		yes			yes
	no			no
			Is the matrix upper triangular?			f01fpf
					yes
				no
			f01fnf


Is $A^{p}$ required?			Is the condition number of the matrix power required?			f01kef
		yes			yes
	no			no
			Is the Fréchet derivative of the matrix power required?			f01kff
					yes
				no
			f01fqf


$f (A)$ will be computed. Will derivatives of $f$ be supplied by the user?			Is the condition number of the matrix function required?			f01kcf
		yes			yes
	no			no
			f01fmf


Is the condition number of the matrix function required?			f01kbf
		yes
	no
f01flf

NAG Library Chapter Introduction

F01 (matop)Matrix Operations, Including Inversion

▸▿ Contents

1 Scope of the Chapter

2 Background to the Problems

2.1 Matrix Inversion

2.2 Matrix Factorizations

2.3 Matrix Arithmetic and Manipulation

2.4 Matrix Functions

3 Recommendations on Choice and Use of Available Routines

3.1 Matrix Inversion

3.2 Matrix Factorizations

3.3 Matrix Arithmetic and Manipulation

3.3.1 NAG Names and LAPACK Names

3.4 Matrix Functions

4 Decision Trees

Tree 1

Tree 2: Inverse of a real n by n matrix of full rank

Tree 3: Inverse of a complex n by n matrix of full rank

Tree 4: Pseudo-inverses

Tree 5: Matrix functions fA of an n by n real matrix A

Tree 6: Matrix functions fA of an n by n complex matrix A

5 Functionality Index

6 Auxiliary Routines Associated with Library Routine Arguments

7 Routines Withdrawn or Scheduled for Withdrawal

8 References

F01 (matop)
Matrix Operations, Including Inversion

1

Scope of the Chapter

2

Background to the Problems

2.1

Matrix Inversion

2.2

Matrix Factorizations

2.3

Matrix Arithmetic and Manipulation

2.4

Matrix Functions

3

Recommendations on Choice and Use of Available Routines

3.1

Matrix Inversion

3.2

Matrix Factorizations

3.3

Matrix Arithmetic and Manipulation

3.3.1

NAG Names and LAPACK Names

3.4

Matrix Functions

4

Decision Trees

Tree 5: Matrix functions $f (A)$ of an n by n real matrix $A$

Tree 6: Matrix functions $f (A)$ of an n by n complex matrix $A$

5

Functionality Index

6

Auxiliary Routines Associated with Library Routine Arguments

7

Routines Withdrawn or Scheduled for Withdrawal

8

References