full-rank representations of outer inverses based on the qr decomposition

Applied Mathematics and Computation 218 (2012) 10321–10333

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Full-rank representations of outer inverses based on the QR decomposition

Predrag S. Stanimirovic a,⇑,1, Dimitrios Pappas b, Vasilios N. Katsikis c, Ivan P. Stanimirovic a,1

a University of Niš, Faculty of Sciences and Mathematics, Višegradska 33, 18000 Niš, Serbiab Department of Statistics, Athens University of Economics and Business, 76 Patission Str, 10434 Athens, Greecec Technological Education Institute of Piraeus, Petrou Ralli & Thivon 250, 12244 Aigaleo, Athens, Greece

a r t i c l e i n f o

Keywords:Generalized inverseOuter inverseQR factorizationFull-rank representation

0096-3003/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.amc.2012.04.011

⇑ Corresponding author.E-mail addresses: [email protected] (P.S. Stanim

(I.P. Stanimirovic).1 The authors gratefully acknowledge support from

a b s t r a c t

An efficient algorithm for computing Að2ÞT;S inverses of a given constant matrix A, based onthe QR decomposition of an appropriate matrix W, is presented. Correlations betweenthe derived representation of outer inverses and corresponding general representationbased on arbitrary full-rank factorization are derived. In particular cases we derive repre-sentations of f2;4g and f2;3g-inverses. Numerical examples on different test matrices(dense or sparse) are presented as well as the comparison with several well-known meth-ods for computing the Moore–Penrose inverse and the Drazin inverse.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Using the usual notation, by Cm�nr we denote the set of all complex m� n matrices of rank r, and by I we denote the unit

matrix of an appropriate order. Furthermore A�;RðAÞ; rankðAÞ and NðAÞ denote the conjugate transpose, the range, the rankand the null space of A 2 Cm�n.

If A 2 Cm�nr ; T is a subspace of Cn of dimension t 6 r and S is a subspace of Cm of dimension m� t, then A has a f2g-inverse

X such that RðXÞ ¼ T and NðXÞ ¼ S if and only if AT � S ¼ Cm, in which case X is unique and it is denoted by Að2ÞT;S. The outergeneralized inverses with prescribed range and null-space are very important in matrix theory. The f2g-inverses have appli-cation in the iterative methods for solving the nonlinear equations [1,12] as well as in statistics [8,9]. In particular, outerinverses play an important role in stable approximations of ill-posed problems and in linear and nonlinear problems involv-ing rank-deficient generalized inverses [11,22]. On the other hand, it is well known that the Moore–Penrose inverse and theweighted Moore–Penrose inverse Ay;AyM;N , the Drazin and the group inverse AD

;A#, as well as the Bott–Duffin inverse Að�1ÞðLÞ

and the generalized Bott–Duffin inverse AðyÞðLÞ can be presented by a unified approach, as generalized inverses Að2ÞT;S for appro-priate choice of matrices T and S. For example, the next (see [1,14,20]) is valid for a rectangular matrix A:

Ay ¼ Að2ÞRðA�Þ;NðA�Þ; AyM;N ¼ Að2ÞRðA]Þ;NðA]Þ; ð1:1Þ

where M;N are positive definite matrices of appropriate orders and A] ¼ N�1A�M. For a given square matrix A the next iden-tities (see [1–3,20]) are satisfied:

AD ¼ Að2ÞRðAkÞ;NðAkÞ

; A# ¼ Að2ÞRðAÞ;NðAÞ; ð1:2Þ

. All rights reserved.

irovic), [email protected] (D. Pappas), [email protected] (V.N. Katsikis), [email protected]

the Research Project 174013 of the Serbian Ministry of Science.

http://dx.doi.org/10.1016/j.amc.2012.04.011

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10322 P.S. Stanimirovic et al. / Applied Mathematics and Computation 218 (2012) 10321–10333

where k ¼ indðAÞ. If A is the L-positive semi–definite matrix and L is a subspace of Cn which satisfiesAL� L? ¼ Cn, S ¼ RðPLAÞ, then the next identities (see [2,20,21]) are satisfied:

Að�1ÞðLÞ ¼ Að2Þ

L;L?; AðyÞðLÞ ¼ Að2Þ

S;S?: ð1:3Þ

The representation of the Moore–Penrose inverse Ay which is based on the QR decomposition of the matrix A is known in theliterature (see, for example, [13,20]). Numerical testing of this representation is presented in [13]. The numerical method forcomputing the minimum-norm least-squares solution Ayb to the undetermined system of linear equations Ax ¼ b, based on aQR factorization of A�, is investigated in [7]. This method can be applied in computation of the Moore–Penrose inverse Ay bytaking successive values b ¼ ei, where ei is the ith unit vector, for each i ¼ 1; . . . ; n.

In the present paper we develop a numerical algorithm (which is called the QRATS2 algorithm) for computing Að2ÞT;S in-verses. The algorithm is based on the full-rank representation of an appropriately chosen matrix W arising from its QRdecomposition. Numerical algorithms are implemented in the numerical computing environment Matlab.

Our paper proceeds as follows. The development of a numerical algorithm for computing Að2ÞT;S inverses which is basedon the QR decomposition of an appropriately chosen matrix W is discussed in Section 2. Numerical examples on varioustest matrices and comparison with known methods for computing the Moore–Penrose inverse and the Drazin inverse arepresented in Section 3. In particular, Section 3 is divided in two subsections; in the first one numerical experiments forthe Moore–Penrose inverse are presented for the cases of random singular matrices, from the Matrix ComputationalToolbox (see [6]) and sparse test matrices from the Matrix Market collection (see [10]), whereas in the second the cor-responding numerical experiments for the Drazin inverse are provided. The conclusions of our work are discussed inSection 4.

2. Representations based on QR decomposition

There exist a number of full-rank representations for outer inverses of prescribed rank as well as for outer inverses withprescribed range and kernel. The following representations from [16] will be useful for the results that follow.

Proposition 2.1 [16]. Let A 2 Cm�nr ; T be a subspace of Cn of dimension s 6 r and let S be a subspace of Cm of dimension m� s. In

addition, suppose that W 2 Cn�m satisfiesRðWÞ ¼ T;NðWÞ ¼ S. Let W ¼ FG be an arbitrary full-rank decomposition of W. If A hasa f2g-inverse Að2ÞT;S, then:

(1) GAF is an invertible matrix;(2) Að2ÞT;S ¼ FðGAFÞ�1G ¼ Að2ÞRðFÞ;NðGÞ.

The representation of Að2ÞT;S inverses given in Proposition 2.1 is theoretically very strong and general. This representation isused in the same paper to give the determinantal representation of Að2ÞT;S. Determinantal representation is not an efficientnumerical procedure for the calculation of generalized inverses. In addition, the authors of the paper [16] do not considerdifferent possibilities regarding the selection of appropriate full-rank factorizations.

The present paper has two main goals. Our first intention is to develop an efficient method for calculating outer inversesusing its general full-rank representation. In addition, our goal is to derive analogous algorithms for calculating f2;4g andf2;3g-inverses, as two particular subsets of outer inverses.

According to known representations from [1,15–18] we state the next additional representations with respect to (1.1)–(1.3). These representations characterize the classes of f2g; f2;4g and f2;3g generalized inverses with prescribed rank.

Proposition 2.2. Let A 2 Cm�nr be an arbitrary matrix, 0 < s 6 r. The following general representations for some classes of

generalized inverses are valid:

ðaÞ Af2gs ¼ fAð2ÞRðFÞ;NðGÞ ¼ FðGAFÞ�1GjF 2 Cn�s; G 2 Cs�m; rankðGAFÞ ¼ sg;

ðbÞ Af2;4gs ¼ Að2;4ÞRððGAÞ�Þ;NðGÞ ¼ ðGAÞ� GAðGAÞ�ð Þ�1GjG 2 Cs�ms

n o

¼ ðGAÞyGjGA 2 Cs�ns

n o;

ðcÞ Af2;3gs ¼ Að2;3ÞRðFÞ;NððAFÞ�Þ ¼ F ðAFÞ�AFð Þ�1ðAFÞ�jF 2 Cn�ss

n o

¼ FðAFÞyjAF 2 Cm�ss

n o;

ðdÞ Af1;2g ¼ Af2gr :

In the following statement we derive a representation of outer inverses with prescribed rank, range and null space. Therepresentation is based on the QR decomposition defined as in Theorem 3.3.11 from [19]. The analogous QR decompositionfor complex matrices is used from [5].

P.S. Stanimirovic et al. / Applied Mathematics and Computation 218 (2012) 10321–10333 10323

Lemma 2.1. Assume that the matrix A 2 Cm�nr is given. Let us consider an arbitrary matrix W 2 Cn�m

s ; s 6 r. Suppose that theQR factorization of W is of the form

WP ¼ QR; ð2:1Þ

where P is an m�m permutation matrix, Q 2 Cn�n;Q �Q ¼ In and R 2 Cn�ms is an upper trapezoidal matrix. Assume that P is chosen

so that Q and R can be partitioned as

Q ¼ Q 1 Q 2½ �; R ¼R11 R12

O O

� �¼

R1

O

� �; ð2:2Þ

where Q1 consists of the first s columns of the matrix Q and R11 2 Cs�s is nonsingular.If A has a f2g-inverse Að2ÞRðWÞ;NðWÞ, then:

(a) R1P�AQ 1 is an invertible matrix;

(b) Að2ÞRðWÞ;NðWÞ ¼ Q 1ðR1P�AQ 1Þ�1R1P�;

(c) Að2ÞRðWÞ;NðWÞ ¼ Að2ÞRðQ1Þ;NðR1P�Þ;

(d) Að2ÞRðWÞ;NðWÞ ¼ Q 1ðQ �1WAQ1Þ�1Q �1W;

(e) Að2ÞRðWÞ;NðWÞ 2 Af2gs.

Proof. (a) According to the assumptions we have

W ¼ QRP�: ð2:3Þ

It is clear that the nontrivial, or ‘skinny’, part of the QR decomposition (2.1), defined by

W ¼ Q 1ðR1P�Þ ð2:4Þ

is a full-rank factorization of W (see also [20]). Since A has a f2g-inverse Að2ÞRðWÞ;NðWÞ, according to part (1) of Proposition 2.1 weconclude that R1P�AQ1 is invertible.

(b), (c) According to part (2) of Proposition 2.1, it is not difficult to derive the representation

Að2ÞRðWÞ;NðWÞ ¼ Q 1ðR1P�AQ 1Þ�1R1P�;

of the outer inverse Að2ÞRðWÞ;NðWÞ with prescribed range RðWÞ ¼ RðQ1Þ and the null space NðWÞ ¼ N ðR1P�Þ.(d) Moreover, from (2.4) we have

R1P� ¼ Q �1W

and

Q 1ðR1P�AQ1Þ�1R1P� ¼ Q 1ðQ �1WAQ1Þ�1Q �1W:

(e) This part of the proof follows from Proposition 2.2. h

Using the results of Proposition 2.2 and taking into account (1.1)–(1.3), we get the following full-rank representations forvarious outer generalized inverses:

Corollary 2.1. For a given matrix A 2 Cm�nr and a selected matrix W 2 Cn�m

s ; s 6 r, with the full-rank representation (2.4) arisingfrom its QR decomposition (2.1), the following statements are valid:

Að2ÞRðQ1Þ;NðR1P�Þ ¼

Ay; W ¼ A�;

AyM;N; W ¼ A];

A#; W ¼ A;

AD; W ¼ Ak

; k P indðAÞ;Að�1ÞðLÞ ; RðWÞ ¼ L; NðWÞ ¼ L?;

AðyÞðLÞ; RðWÞ ¼ S; NðWÞ ¼ S?:

8>>>>>>>>>><>>>>>>>>>>:

ð2:5Þ

Full-rank factorizations W ¼ FG which satisfy F�F ¼ I are called orthogonal [14]. Therefore, the full-rank representation (2.4)is orthogonal.

In the following statement we show that the orthogonal full-rank representation based on the QR decomposition of Wproduces the same Að2ÞT;S inverse as the representation from Proposition 2.1.

Corollary 2.2. Assume that all assumptions of Lemma 2.1 are valid and W ¼ FG is an arbitrary full-rank factorization of W. Then

Q 1ðR1P�AQ1Þ�1R1P� ¼ FðGAFÞ�1G: ð2:6Þ


Proof. Using uniqueness of the generalized inverse with prescribed range and null space and applying part (b) of Lemma 2.1and Proposition 2.1, part (2), we conclude

Q1ðR1P�AQ 1Þ�1R1P� ¼ Að2ÞRðWÞ;NðWÞ ¼ Að2ÞRðFÞ;NðGÞ ¼ FðGAFÞ�1G;

which completes the proof. h

In the following lemma we show that the representation of the outer inverse Að2ÞRðWÞ;NðWÞ, defined in Proposition 2.1, isinvariant with respect to the choice of the full-rank factorization of W. In this way, we obtain a generalization ofCorollary 2.2.

Lemma 2.2. Let the assumptions of Lemma 2.1 be satisfied. If W ¼ F1G1 and W ¼ F2G2 are two different full-rank factorizationsof the matrix W, we have

F1ðG1AF1Þ�1G1 ¼ F2ðG2AF2Þ�1G2 ¼ Að2ÞRðWÞ;NðWÞ:

Proof. According to Theorem 2 from [14], there exists an invertible s� s matrix D such that

F2 ¼ F1D; G2 ¼ D�1G1:

This relationship implies the following

F2ðG2AF2Þ�1G2 ¼ F1D D�1G1AF1D� ��1

D�1G1:

Since D and G1AF1 are invertible, the reverse order law for D�1G1AF1D� ��1

is satisfied, so that

F2ðG2AF2Þ�1G2 ¼ F1DD�1ðG1AF1Þ�1DD�1G1 ¼ F1ðG1AF1Þ�1G1:

Now it is not difficult to complete the proof. h

We finally find a correlation between the general representation given in the paper [16] and the representation presentedin Lemma 2.1.

Corollary 2.3. Assume that the conditions of Lemma 2.1 are satisfied. Let W ¼ FG be an arbitrary full-rank representation of Wsatisfying conditions from Proposition 2.1. If (2.4) is a full-rank factorization of W, we have

F ¼ Q1D; G ¼ D�1R1P�; ð2:7Þ

where the matrix D is equal to D ¼ Q �1F.

Example 2.1. Let us consider the matrix

A ¼

1 2 3 4 11 3 4 6 22 3 4 5 33 4 5 6 44 5 6 7 66 6 7 7 8

2666666664

3777777775

ð2:8Þ

and choose the matrices

F ¼

3 �2�1 10 00 00 0

26666664

37777775; G ¼

1 0 0 0 0 00 1 0 0 0 0

� �: ð2:9Þ

Assume that the matrix W is defined by W ¼ FG. The matrices Q 1;R1 and P which define the nontrivial part of the QR decom-position (2.1) of WP are given by


fQ 1;R1; Pg ¼

3ffiffiffiffi10p 1ffiffiffiffi

10p

� 1ffiffiffiffi10p 3ffiffiffiffi

10p

0 00 00 0

26666664

37777775;

ffiffiffiffiffiffi10p

� 7ffiffiffiffi10p 0 0 0 0

0 1ffiffiffiffi10p 0 0 0 0

" #; I6

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;:

The representations given in statements (b) and (c) of Lemma 2.1 give the following f2g-inverse of A:

Að2ÞRðWÞ;NðWÞ ¼

3 �2 0 0 0 0�1 1 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

26666664

37777775:

Although W ¼ FG and W ¼ Q 1ðR1P�Þ are two different full-rank representations of W, in accordance with Corollary 2.2 wehave

Að2ÞRðWÞ;NðWÞ ¼ Q 1ðR1P�AQ 1Þ�1R1P� ¼ FðGAFÞ�1G:

Let us denote by IðnÞ the complexity of the algorithm for inverting a given n� n matrix (as in [4]). Also by AðnÞwe denotethe complexity of the addition/subtraction of two n� n matrices and by Mðm;n; kÞ the complexity of multiplying m� n ma-trix with n� k matrix. The simpler notation MðnÞ (taken from [4]) is used instead of Mðn;n;nÞ.

Remark 2.1. The representation given in the statement (d) of Lemma 2.1 requires Mðs;n;mÞ flops to form the matrix productQ �1W . On the other hand, the representation defined in the statement (b) of Lemma 2.1 requires Mðs;m;mÞ flops to form thematrix product R1P�. Therefore, the representation (d) is more advantageous than (b) in the case when n < m. In the oppositecase m < n the representation (b) is more appropriate with respect to the representation (d).

An efficient method for computing Að2ÞRðWÞ;NðWÞ is defined as follows. In the case m < n, considering the complexity of matrixmultiplications, it is more suitable to solve the set of equations

R1P�AQ1X ¼ R1P�: ð2:10Þ

In the case n < m, it is more efficient to solve the set of equations

Q �1WAQ1X ¼ Q �1W: ð2:11Þ

When the matrix X is generated, it is necessary to compute the matrix product

Að2ÞRðQ1Þ;NðR1P�Þ ¼ Að2ÞRðQ1Þ;NðQ�1WÞ ¼ Q 1X: ð2:12Þ

Corollary 2.4. Let A 2 Cm�nr be the given matrix, s 6 r be a given integer and the matrices F, G are chosen as in Proposition 2.2.

(a) If (2.4) is a full-rank factorization of W ¼ ðGAÞ�G 2 Cn�ms the following is satisfied:

Að2;4ÞRðQ1Þ;NðR1P�Þ ¼ Q 1ðR1P�AQ 1Þ�1R1P� ¼ ðGAÞyG 2 Af2;4gs: ð2:13Þ

(b) If (2.4) is a full-rank factorization of W ¼ FðAFÞ� 2 Cn�ms the following holds:

Að2;3ÞRðQ1Þ;NðR1P�Þ ¼ Q 1ðR1P�AQ 1Þ�1R1P� ¼ FðAFÞy 2 Af2;3gs: ð2:14Þ

Proof

(a) In this case, according to Corollary 2.2 and Proposition 2.2 we have

Q 1ðR1P�AQ1Þ�1R1P� ¼ ðGAÞ�ðGAðGAÞ�Þ�1G ¼ ðGAÞyG:

(b) This part of the proof can be verified in a similar way. h


Example 2.2. Let us consider the matrix A defined in (2.8) and choose matrices F;G defined in (2.9).

(i) The full-rank representation (2.4) of W ¼ ðGAÞ�G is given by

fQ1;R1; Pg ¼

1ffiffiffiffi31p �2

ffiffiffiffi7

93

q2ffiffiffiffi31p

ffiffiffiffiffiffi3

217

q3ffiffiffiffi31p � 11ffiffiffiffiffiffi

651p

4ffiffiffiffi31p 2

ffiffiffiffiffiffi3

217

q1ffiffiffiffi31p 17ffiffiffiffiffiffi

651p

266666666664

377777777775;


45ffiffiffiffi31p 0 0 0 0

0ffiffiffiffi2131

q0 0 0 0

24

35; I6

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;:

The representation (b) from Lemma 2.1 produces the following f2;4g-inverse of A:

Að2;4ÞRðQ1Þ;NðR1P�Þ ¼ Q 1ðR1P�AQ1Þ�1R1P� ¼

1 � 23 0 0 0 0

� 17

17 0 0 0 0

67 � 11

21 0 0 0 0� 2

727 0 0 0 0

� 87

1721 0 0 0 0

26666664

37777775¼ ðGAÞyG ¼ Að2;4ÞRððGAÞ�Þ;NðGÞ:

(ii) Let us choose the matrix W as W ¼ FðAFÞ�. The full-rank representation (2.4) of W is given by

fQ1;R1; Pg

3ffiffiffiffi10p 1ffiffiffiffi

10p


10p

0 00 00 0

26666664

37777775;



10p 32

ffiffi25

q91ffiffiffiffi10p 81

ffiffi25

q0 1ffiffiffiffi

10p � 1ffiffiffiffi

10p �

ffiffi25

q� 3ffiffiffiffi

10p �3

ffiffi25

q264

375; I6

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;:

The representation from Lemma 2.1, part (b) gives the following f2;3g-inverse of A:

Að2;3ÞRðQ1Þ;NðR1P�Þ ¼ Q 1ðR1P�AQ1Þ�1R1P� ¼

� 59392 � 69

196 � 39392 � 19

3921

3921549

55392

61196

43392

31392

19392 � 9

49

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

26666664

37777775¼ FðAFÞy ¼ ARðFÞ;NððAFÞ�Þ:

In the sequel we state the algorithm for generating Að2ÞT;S inverses of a given matrix A.

Algorithm 2.1. Computing the Að2ÞT;S inverse of the matrix A using the QR decomposition of W. (Algorithm QRATS2)

Require: The matrix A of dimensions m� n and of rank r.1: Choose arbitrary but fixed n�m matrix W of rank s 6 r2: Compute the QR decomposition of the matrix W in the form (2.1).3: Generate the full-rank decomposition of the matrix W as in (2.4).4: Solve the matrix equation of the form (2.10) in the case m < n or the matrix Eqs. (2.11) in the case n < m.

5: Compute the output Að2ÞRðQ1Þ;NðR1P�Þ ¼ Að2ÞRðQ1Þ;NðQ�1WÞ as in (2.12).

2.1. Complexity of algorithms QRATS2 and qrginv

Algorithm QRATS2 uses the QR decomposition of the matrix W of the form (2.4) and then computes Að2ÞT;S inverse accordingto the statement of part (b) in Lemma 2.1. Therefore, its computational complexity is equal to

EðQRATS2ðWÞÞ ¼Mðs;m;mÞ þMðs;m;nÞ þMðs;n; sÞ þ IðsÞ þMðn; s; sÞ þMðn; s;mÞ: ð2:15Þ

In Table 1 we describe the computational complexity required to compute the outer inverse X ¼ Q 1ðR1P�AQ 1Þ�1R1P�.It is well-known that matrix inversion is equivalent to matrix multiplication. More precisely, the ordinary inverse of any

real nonsingular n� n matrix can be computed in time IðnÞ ¼ OðMðnÞÞ [4]. The notation Oðf ðnÞÞ is explained, for example, in[4]. In our implementation we use the usual matrix multiplication methods, so that Mðm;n; kÞ ¼ m � n � k. This implies,according to (2.15)

Table 1Computational complexity of X ¼ Q 1ðR1P�AQ 1Þ�1R1P� .

Expression Complexity

K1 ¼ R1P� Mðs;m;mÞK2 ¼ K1A Mðs;m;mÞ þMðs;m;nÞK3 ¼ K2Q1 Mðs;m;mÞ þMðs;m;nÞ þMðs;n; sÞK4 ¼ ðK3Þ�1 Mðs;m;mÞ þMðs;m;nÞ þMðs;n; sÞ þ IðsÞK5 ¼ Q1K4 Mðs;m;mÞ þMðs;m;nÞ þMðs;n; sÞ þ IðsÞ þMðn; s; sÞX ¼ K5K1 Mðs;m;mÞ þMðs;m;nÞ þMðs;n; sÞ þ IðsÞ þMðn; s; sÞ þMðn; s;mÞ


EðQRATS2ðWÞÞ ¼ 2s2nþ smðmþ 2nÞ þ IðsÞ � 2s2nþ smðmþ 2nÞ þ s3: ð2:16Þ

In the particular case W ¼ A�, Algorithm QRATS2 uses the QR decomposition of the matrix A� of the form (2.4) and then com-putes Ay. On the other hand, Algorithm qrginv starts from the QR decomposition of the matrix A of the form

A ¼ QARAP�A

and computes the Moore–Penrose inverse

Ay ¼ PARyAQ �A:

Its computational complexity is given by

EðqrginvÞ ¼ pinvðr; nÞ þMðn;n; rÞ þMðn; r;mÞ ¼ rnðmþ nÞ þ pinvðr;nÞ; ð2:17Þ

where pinvðr;nÞ denotes the computational complexity of a rectangular r � n real matrix.In the particular case W ¼ A� of Algorithm QRATS2 its computational complexity is

EðQRATS2ðA�ÞÞ ¼ r2ðmþ 2nÞ þ 2rmnþ pinvðr; rÞ: ð2:18Þ

In comparison of EðqrginvÞ and EðQRATS2ðA�ÞÞ the fact pinvðr; rÞ 6 pinvðr;nÞ must be taken into account.

2.2. From QRATS2 to qrginv and vice versa

Suppose that AP ¼ QR, is a QR decomposition defined by the matrix Q with orthonormal columns and R ¼ R1

0

� �, with R1

full row rank (as in [19], Theorem 3.3.11). Then AP ¼ Q1R1, after cutting rows and columns by taking into account the rank of

A. According to [13] (comment after Theorem 4), it holds

Ay ¼ PRyQ � ¼ P Ry1 0� �

Q � ¼ PRy1Q �1: ð2:19Þ

In the present article we define two representations of the Moore–Penrose inverse. The first representation is defined as theparticular case W ¼ A� of Algorithm QRATS2. More precisely, it starts from the QR decomposition A� ¼ Q 1ðR1P�Þ of the matrixA� and then computes the Moore–Penrose inverse according to the part (b) of Lemma 2.1.

The second representation of the Moore–Penrose inverse is based on the decomposition of the matrix A�, obtained fromthe QR decomposition of the matrix A. If AP ¼ Q1R1; Q �1Q 1 ¼ I then A� ¼ ðPR�1ÞQ

�1 is a full-rank factorization of A�. From Lem-

ma 2.1 we obtain

QRATS2ðA�Þ ¼ PR�1 Q �1APR�1 �1Q �1: ð2:20Þ

It is not difficult to verify that the representation (2.20) produces the same result as the representation (2.19). Indeed, using

QRATS2ðA�Þ ¼ PR�1 Q �1APR�1 �1Q �1 ¼ PR�1 Q �1Q1R1P�PR�1

�1Q �1 ¼ PR�1ðR1R�1Þ�1Q �1

and taking into account that R1 is of full row rank, it follows that QRATS2ðA�Þ ¼ PRy1Q �1, which is exactly (2.19).

3. Numerical experiments

3.1. Numerical experiments for the Moore–Penrose inverse

In what follows, we make use of the high-level language Matlab both for calculations, as well as for testing the reliabilityof the obtained results. All the numerical tasks have been performed by using Matlab R2009a environment on an Intel(R)Pentium(R) Dual CPU T2310 1.46 GHz 1.47 GHz 32-bit system with 2 GB of RAM memory running on the Windows VistaHome Premium Operating System.

We perform numerical experiments comparing the algorithm QRATS2 with the algorithm qrginv from [13] and with theiterative methods presented by Wei and Wu in [21], in order to compute the Moore–Penrose inverse of a singular matrix A.


In the work of Wei–Wu, [21], several iterative methods are presented, and after testing them with different collections ofmatrices we concluded that the most reliable from the point of both accuracy and elapsed time is the one presented inEq. (3.8) in [21]. Therefore only this method from [21] will be included in our numerical tests.

Testing of QRATS2, qrginv and the Wei–Wu method was performed separately for random singular matrices and for sin-gular test matrices with ‘‘large’’ condition number from the Matrix Computation Toolbox (see [6]). We also tested the pro-posed method on sparse matrices from the Matrix Market collection (see [10]), and we obtained very fast and accurateresults in all cases.

Finally, we present numerical results for the Drazin inverse case i.e., the case W ¼ Ak, where k is the index of matrix A.Note that, in this section, the stroke ‘–’ means a long processor time required for the computation. The tolerance for the iter-ative method (3.8) from [21] is set equal to 10�6. The implementation of the QRATS2 method in Matlab, as well as a Matlabfunction for the calculation of the matrix index are provided in the appendix. The accuracy of the results is verified using thefollowing four relations which characterize the Moore–Penrose inverse:

Table 2Error an

Rank

28n1

29n1

210n

211n

Table 3Error an

Meth

QRAqrginWei,

Table 4Error an

Meth

QRAqrginWei,

AAyA ¼ A; AyAAy ¼ Ay; ðAAyÞ� ¼ AAy; ðAyAÞ� ¼ AyA:

3.1.1. Random singular test matricesWe are comparing the performance of the proposed method QRATS2 with the algorithm qrginv from [13] and the method

proposed by Eq. (3.8) in [21], on a series of random singular matrices with rank 2n; n ¼ 8;9;10;11. In addition, the accuracyof the results is examined with the matrix 2-norm in error matrices corresponding to the four properties characterizing theMoore–Penrose inverse.

In Table 2, the accuracy of the proposed methods is tested based on the 2-norm errors. Let us recall that Rank (resp. Cond)column contains the rank (rep. the condition number) of each tested matrix. It is evident that the proposed method QRATS2produced a reliable approximation in all the tests that were conducted. The proposed method allows us for fast and accuratecomputations, hence it can be a reliable alternative for computing the Moore–Penrose inverse.

3.1.2. Singular test matrices from the Matrix Computation ToolboxThe condition numbers of these matrices range from order 1015 to 10135. For comparative purpose we also apply as in the

previous section, the proposed QRATS2 algorithm, the qrginv function and the Wei–Wu method. Since these matrices are of

d computational time results; random singular matrices.

nCond Method Time kAAyA� Ak2 kAyAAy � Ayk2 kAAy � ðAAyÞ�k2 kAyA� ðAyAÞ�k2

.14003e+04 QRATS2 0.1796 1.1969e�12 3.8694e�12 3.1776e�12 2.2822e�12

qrginv 0.1929 2.6305e�12 6.4950e�12 4.7533e�12 5.7056e�13Wei,Wu 5.4895 4.4502e�12 4.6559e�08 2.8868e�13 1.1914e�11

.2965e+04 QRATS2 1.3490 6.7212e�12 7.1291e�11 5.9296e�12 1.2068e�11


6.3175e+05 QRATS2 10.0298 6.5701e�11 1.0135e�09 1.4507e�10 9.7971e�11


8.8828e+04 QRATS2 76.1483 5.6069e�11 2.1023e�10 5.7148e�11 9.1536e�11

qrginv 64.5475 3.2243e�11 9.3141e�12 2.6953e�11 2.3243e�12Wei,Wu -

d computational time results for chow (Rank = 199, Cond = 8.01849e+135).

od Time kAAyA� Ak2 kAyAAy � Ayk2 kAAy � ðAAyÞ�k2 kAyA� ðAyAÞ�k2

TS2 0.0284 9.4187e�13 2.0040e�13 1.8662e�13 1.3462e�12v 0.0695 4.3737e�13 1.1428e�13 6.0657e�13 1.4614e�13

Wu 1.6319 3.9456e�13 5.8480e�13 8.9149e�15 1.5847e�14

d computational time results for cycol (Rank = 50, Cond = 2.05e+48).

od Time kAAyA� Ak2 kAyAAy � Ayk2 kAAy � ðAAyÞ�k2 kAyA� ðAyAÞ�k2

TS2 0.0154 3.5703e�14 8.3310e�17 1.7152e�15 3.8605e�15v 0.0479 3.3422e�14 6.3095e�17 1.5658e�15 1.2189e�15

Wu 0.5201 5.1488e�07 2.3601e�09 2.4669e�15 1.0209e�15

Table 5Error and computational time results for gearmat (Rank = 199, Cond = 3.504e+17).

Method Time kAAyA� Ak2 kAyAAy � Ayk2 kAAy � ðAAyÞ�k2 kAyA� ðAyAÞ�k2

QRATS2 0.0172 4.1331e�15 4.0777e�13 1.0614e�14 1.2640e�13qrginv 0.0683 3.3368e�15 1.6225e�13 9.5534e�14 2.1089e�14Wei,Wu 1.2365 4.3255e�13 4.3828e�10 4.8506e�15 5.3844e�15

Table 6Error and computational time results for kahan (Rank = 199, Cond = 2.30018e+24).


QRATS2 0.0199 3.1363e�14 1.1835e�09 6.5937e�10 3.9813e�14qrginv 0.0535 2.1280e�05 1.8366e�09 0.6923 7.5374e�15Wei,Wu 4.0873 7.6304e�15 2.9332e�10 3.6749e�10 3.2342e�14

Table 7Error and computational time results for lotkin (Rank = 19, Cond = 8.97733e+21).


QRATS2 0.0912 8.5749e�06 9.5852e�08 3.1702e�11 0.2384qrginv 0.0461 8.3470e�06 8.6095e�09 0.0463 1.1078e�11Wei,Wu –

Table 8Error and computational time results for prolate (Rank = 117, Cond = 5.61627e+17).



Table 9Error and computational time results for hilb (Rank = 20, Cond = 1.17164e+19).



Table 10Error and computational time results for magic (Rank = 3, Cond = 4.92358e+19).


QRATS2 0.0856 1.0460e�08 2.4776e�19 7.6620e�14 2.9086e�13qrginv 0.0455 1.1003e�08 5.6690e�19 4.0822e�13 3.0107e�14Wei,Wu 1.0240 2.5179e+06 2.5641e�07 2.5206e�17 1.9203e�17

Table 11Error and computational time results for vand (Rank = 34, Cond = 1.16262e+20).




Table 12Error and computational time results for Matrix Market sparse matrices.

Matrix Method Time kAAyA� Ak2 kAyAAy � Ayk2 kAAy � ðAAyÞ�k2 kAyA� ðAyAÞ�k2

WELL1033_Z QRATS2 0.0090 7.6971e�13 1.7900e�10 2.2215e�10 4.2044e�11(m ¼ 1033) qrginv 0.0098 8.7761e�13 2.0632e�10 7.3052e�11 1.6903e�12WELL1850_Z QRATS2 0.0812 1.6485e�12 1.2869e�10 1.8413e�09 6.4710e�11(m ¼ 1850) qrginv 0.0582 1.2844e�12 1.0488e�10 7.2368e�11 2.0119e�12ILCC1850_Z QRATS2 0.0159 1.7551e�11 1.0295e�7 1.6967e�08 5.0640e�09(m ¼ 1850) qrginv 0.0570 9.9285e�12 3.9597e�08 5.1084e�09 6.9214e�11GR-30–30_Z QRATS2 0.1231 4.7019e�11 2.2368e�10 8.1965e�12 8.5861e�10(m ¼ 900) qrginv 0.0487 1.2309e�11 3.8569e�10 2.4849e�10 7.3320e�12WATT1_Z QRATS2 0.0022 1.8520e�05 5.0877e�16 3.2254e�19 5.2566e�13(m ¼ 1856) qrginv 0.0127 3.1237e�13 0.1252 3.4532e�09 0.9173e�06

Note: In parenthesis is denoted the row size (m) of each matrix.


relatively small size and so as to measure the time needed for each algorithm to compute the Moore–Penrose inverseaccurately, each algorithm runs 100 distinct times. The reported time is the mean time over these 100 repetitions. The errorsare presented in Tables 3–11.

It seems interesting to observe the complementarity of algorithms QRATS2 and qrginv in Tables 7, 8, 9, 11. More precisely,results from Algorithm QRATS2 act as f1;2;3g inverses while qrginv algorithm produces the results acting as f1;2;4g in-verses. Algorithm QRATS2 uses the QR decomposition of the matrix A� while Algorithm qrginv starts from the QR decompo-sition of the matrix A. This fact may be the cause for the difference in the convergence of Algorithm QRATS2 in the caseW ¼ A� and Algorithm qrginv. If the Eq. (2.20) is used, then there are no complementary results in Tables 7, 8, 9, 11.

3.1.3. Matrix market sparse test matricesIn this section we test the proposed algorithm on sparse matrices from the Matrix Market collection (see [10]). We follow

the same method and the same matrices as in [13], while the matrices are taken as rank deficient: A Z ¼ ½A Z�, where A isone of the chosen matrices, Z is a zero matrix of order m� 100 and m takes values shown in Table 12. Since the Wei–Wuiterative method did not produce results within reasonable CPU times, it is omitted from this comparison.

3.2. Numerical experiments for the Drazin inverse

We will present numerical experiments concerning the Drazin inverse AD of a square matrix A when the proposed algo-rithm for the Að2ÞT;S is used. In this case, we have that W ¼ Ak, where k is the index of the matrix A. The index of a square matrixA is the smallest nonnegative integer k such that the statement rankðAkÞ ¼ rankðAkþ1Þ is true.

In order to check the accuracy of the results, we use the following three relations from the definition of the Drazininverse:

ADA ¼ AAD; ADAAD ¼ AD; Akþ1AD ¼ Ak:

At first we present a low-dimensional example for the calculation of the Drazin inverse of a given square matrix by using theproposed method QRATS2.

Example 3.1. Consider the singular matrix

A ¼2 0 00 1 10 �1 �1

264

375:

Then, the index of A is indðAÞ ¼ 2 and

W ¼ A2 ¼4 0 00 0 00 0 0

264

375:

The full-rank QR decomposition of the matrix W is determined as

W ¼ Q 1R1P� ¼100

264

375 4 0 0½ �

1 0 00 1 00 0 1

264

375:

Therefore, according to part (b) of Lemma 2.1 and Corollary 2.1, we have

AD ¼ Q 1ðR1P�AQ 1Þ�1R1P� ¼

12 0 00 0 00 0 0

264

375:

Table 13Error and computational time results for the Drazin inverse.

Matrix Index Rank Method Time kAAD � ADAk2 kADAAD � ADk2 kAkþ1AD � Akk2

CYCOL 1 50 QRATS2 0.2035 2.2280e�12 2.9286e�12 6.3953e�12Wei,Wu – – – –

GEARMAT 2 199 QRATS2 0.3617 1.8144e�10 3.9470e�10 6.3876e�13Wei,Wu – – – –

KAHAN 25 199 QRATS2 1.1954 5.5349e+03 9.7653e�07 2.1696e�10Wei,Wu – – – –

LOTKIN 5 19 QRATS2 0.4343 3.2649e�04 1.1547e�13 5.2916e�06Wei,Wu – – – –

PROLATE 5 117 QRATS2 0.3693 9.2475e�04 1.6430e�14 5.4875e�08Wei,Wu – – – –

HILB 7 20 QRATS2 0.5537 5.2832e�05 2.0200e�14 1.9039e�08Wei,Wu – – – –

MAGIC 1 3 QRATS2 0.1868 4.1704e�13 1.7757e�17 5.7818e�07Wei,Wu 0.1621 6.2739e�18 7.3154e�08 2.3074e+06

VAND 5 34 QRATS2 0.4404 2.2305e+05 4.6483 2.3188e�05Wei,Wu – – – –

Table 14Error and computational time results; random singular matrices.

Index Rank Time kAAD � ADAk2 kADAAD � ADk2 kAkþ1AD � Akk2

1 28 0.8124 1.9117e�10 2.4519e�09 3.5369e�10

1 29 2.4586 5.8145e�11 1.4301e�10 2.2431e�10

1 210 13.4005 2.2166e�09 1.6988e�08 8.4017e�09

1 211 187.3001 2.2031e�09 7.4148e�09 8.4040e�09


In the sequel we present two tables comparing the errors and the computational time of the proposed method to the methodpresented in [21]. We use, once more, a set of eight singular test matrices of size 200� 200 obtained from the function ma-

trix in the Matrix Computation Toolbox (Table 13) and a series of random singular matrices with rank 2n; n ¼ 8;9;10;11(Table 14).

As we can clearly see in Table 13, the iterative method of Wei–Wu ([21]) does not converge in most cases, therefore itcannot produce a numerical answer. On the contrary, the Að2ÞT;S algorithm is giving very accurate answers in most cases. More-over, the elapsed computational time is very small.

The Wei–Wu iterative method did not produce results within reasonable CPU times for the case of random singular matri-ces, so it is omitted from comparisons in Table 14.

4. Conclusions

A numerical algorithm for computing Að2ÞT;S inverses, which generalizes known representation of the Moore–Penrose in-verse Ay from [13], is introduced.

The algorithm from [13] is based on the QR decomposition of the matrix A. Our numerical algorithms is based on the full-rank decomposition, arising from its QR decomposition, of an appropriately chosen matrix W.

We prove that the representation of outer inverses derived in the present paper and corresponding general representationbased on arbitrary full-rank factorization produce identical results. An explicit transitions formulae between these represen-tations is investigated.

Particular rank factorizations which generate f2;4g and f2;3g-inverses are found.The introduced algorithm has proven to be very effective in comparison with other known methods for computing the

Moore–Penrose inverse and the Drazin inverse.

Acknowledgements

The authors gratefully acknowledge anonymous referee’s comments which improved the quality of the paper.

Appendix A

The Matlab implementation of the Algorithm QRATS2 is given below. In the implementation the system (2.10) is solvedusing the Matlab backslash operator. Our results are generated for the case m < n. Similar results can be presented for thecase m > n under a slight modification of the given Matlab code.


The index of a given matrix can be calculated with the following Matlab function:


References

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http://www.ma.man.ac.uk/~higham/mctoolbox

http://www.ma.man.ac.uk/~higham/mctoolbox

http://arxiv.org///math.nist.gov/MatrixMarket

http://arxiv.org///math.nist.gov/MatrixMarket

full-rank representations of outer inverses based on the qr decomposition

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