A Note on Computing the Generalized Inverse A^(2)_{T,S} of a Matrix A

The generalized inverse A T,S (2) of a matrix A is a {2}-inverse of A with the prescribed range T and null space S. A representation for the generalized inverse A T,S (2) has been recently developed with the condition σ (GA| T)⊂(0,∞), where G is a matrix with R(G)=T andN(G)=S. In this note, we remove the above condition. Three types of iterative methods for A T,S (2) are presented if σ(GA|T) is a subset of the open right half-plane and they are extensions of existing computational procedures of A T,S (2), including special cases such as the weighted Moore-Penrose inverse A M,N † and the Drazin inverse AD. Numerical examples are given to illustrate our results.


A NOTE ON COMPUTING THE GENERALIZED INVERSE A
The generalized inverse A (2) T ,S of a matrix A is a {2}-inverse of A with the prescribed range T and null space S. A representation for the generalized inverse A (2) T ,S has been recently developed with the condition σ (GA| T ) ⊂ (0, ∞), where G is a matrix with R(G) = T and N(G) = S.In this note, we remove the above condition.Three types of iterative methods for A (2) T ,S are presented if σ (GA| T ) is a subset of the open right half-plane and they are extensions of existing computational procedures of A 1. Introduction.Given a complex matrix A ∈ C m×n , any matrix X ∈ C n×m satisfying XAX = X is called a {2}-inverse of A. Let T and S be subspaces of C n and C m , respectively.A matrix X ∈ C n×m is called a {2}-inverse of A with the prescribed range T and null space S, denoted by A (2) T ,S , if the following conditions are satisfied: where R(X) is the range of X and N(X) is the null space of X.It is a well-known fact [1] that if dim T = dim S ⊥ ≤ rank(A), then there exists a unique A (2) T ,S if and only if AT ⊕ S = C m .It is obvious from the definition above that AA (2) T ,S = P AT ,S and A (2) T ,S A = P T ,(A * S ⊥ ) ⊥ , where P S 1 ,S 2 is the projector on the subspace S 1 along the subspace S 2 .
There are seven types of important {2}-inverses of A: the Moore-Penrose inverse A † , the weighted Moore-Penrose inverse A † M,N , the W -weighed Drazin inverse A d,w , the Drazin inverse A D , the group inverse A # , the Bott-Duffin inverse A , where N and M are Hermitian positive definite matrices of order n and m, respectively; , where W ∈ C n×m and q = Ind(W A), the index of W A.
S,S ⊥ , where S = R(P L A).
The {2}-inverse has many applications, for example, the application in the iterative methods for solving nonlinear equations [1,9] and the applications to statistics [6,7].In particular, {2}-inverse plays an important role in stable approximations of ill-posed problems and in linear and nonlinear problems involving rank-deficient generalized inverse [8,12].In literature, researchers have proposed many numerical methods for computing A (2) T ,S , see [2,3,11,13,15,16,18].As usual, we denote the spectrum and the spectral radius of A by σ (A) and ρ(A), respectively.The notation • stands for the spectral norm.The following theorem applied in this note is from the theory of semi-iterative method.
Theorem 1.2 (see [5]).Let B ∈ C n×n be a nonsingular matrix and let σ (B) ⊂ Ω, where Ω is a simply connected compact set excluding origin.If a sequence of polynomials {s m (z)} ∞ m=0 uniformly converges to 1/z on Ω, then {s m (B)} converges to B −1 .
In this note, a representation for the generalized inverse A (2) T ,S with a condition σ (GA| T ) ⊂ {z : Re(z) > 0}, where G is a matrix with R(G) = T and N(G) = S is presented in Section 2. Euler-Knopp iterative method and semi-iterative methods for A (2) T ,S with linear convergence are derived in Section 3. Quadratically convergent methods for A (2) T ,S are developed in Section 4. Finally, numerical examples are given to illustrate our results.

Representation.
In this section, we give a representation for the generalized inverse A (2) T ,S , which may be viewed as an application of the classical theory summability to the representation of generalized inverse.Lemma 2.1 (see [13]).Suppose A ∈ C m×n .Let T and S be subspaces of It follows from Lemma 1.1 that the existence of G is assured for each of the common seven types of generalized inverses: A * , N −1 A * M, A(W A) q , A k , A, P L , and P S .Now we are in a position to establish a presentation theorem.Theorem 2.2.Let A, T , S, G, and Ã be as in Lemma 2.1.If σ ( Ã) is contained in a simply connected compact set Ω excluding origin and a polynomial sequence {s m (z)} uniformly converges to 1/z on Ω, then Furthermore, where P is invertible such that P −1 GAP is the -Jordan canonical form of GA and Proof.Assume that σ ( Ã) ⊂ Ω.With applying Theorem 1.2, we get T ,S . (2.5) The error can be written as T ,S . (2.6) Since P is nonsingular such that P −1 GAP is the -Jordan canonical form of GA, it is well known that Thus T ,S P . (2.8) The last inequality is based on the spectrum mapping since s m (z) is a polynomial in z.This completes the proof.
In order to make use of this general error estimate in Theorem 2.2 on specific approximation procedures, it will be convenient to have lower and upper bounds for σ ( Ã).This is given in the next lemma.

Lemma 2.3. Let A, T , S, G, and Ã be as in Lemma 2.1. Then for each
(2.9) Proof.We only show the first inequality since the second is trivial.It follows from Lemma 2.1 that Ind(GA) = 1.Then the Jordan canonical form of GA is where C is invertible.For each λ ∈ σ ( Ã), 1/λ ∈ σ ( Ã−1 ) since Ã is invertible.Consequently, we have which leads to (2.9).This completes the proof.
Remark 2.4.Theorem 2.2 extends the representation of A (2) T ,S in [15] in which σ (GA| T ) ⊂ (0, ∞) is required.The theorem also recovers the representations of A D in [16] and A † M,N in [17] as special cases.

Iterative methods for A (2)
T ,S .In this section, we present applications of Theorem 2.2 and Lemma 2.3 in developing specific computational procedures for the generalized inverse A (2) T ,S and estimating corresponding error bounds.A well-known summability method is called the Euler-Knopp method.A series ∞ m=0 a m is said to be Euler-Knopp summable with parameter α > 0 to the value a if the sequence defined by It follows from Lemma 2.3 that where r 1 = 1/ (GA) # and r 2 = GA .It can be shown with the law of Sines that then σ Ã ⊂ E α .There is always a simply connected compact set Ω such that σ ( Ã) ⊂ Ω ⊂ E α .Hence s m (z) of (3.2) uniformly converges to 1/z on Ω.It follows from Theorem 2.2 that Notice that if A m is the mth partial sum, that is, A m = α m j=0 (I − αGA) j G, then an iteration form for {A m } is given by For an error bound, we note that the sequence of polynomials {s m (z)} satisfies where Actually, by the maximum modular theorem, max z∈F |1 − αz| = max z∈∂F |1 − αz|.We denote four parts of ∂F as follows: With an analogous argument, we have max T ,S , if 0 < α < 2 cos φ/ GA , where φ is given by (3.3).Moreover, the relative error is bounded by (3.16).
We remark that Theorem 3.1 is an extension of corresponding results in [15,16].The procedure of semi-iterative methods [5,10] for solving a linear system can easily be extended to solve If ρ(H) < 1, then a sequence of matrices {X m }, yielded by Moreover, the matrices Y m and the corresponding residual matrices R m are given by where Especially, Ω 1 is either a complex segment [α, β] excluding 1 or a closed ellipse in the left half-plane {z : Re(z) < 1} with foci α and β.Let a sequence of polynomials {p m (z)} given by where T m is the mth Chebyshev polynomial.The semi-iterative method induced by {p m (z)} is the Chebyshev iterative method optimal for ellipse Ω 1 .The corresponding two-step stationary method with the same asymptotically optimal convergence rate is given by where The sequence {Y m } converges asymptotically optimally to A ( T ,S . 4. Quadratically convergent methods.Newton-Raphson method for finding the root 1/z of the function s(w) = w −1 − z is given by w m+1 = w m 2 − zw m , for a suitable w 0 . (4.1) For α > 0, a sequence of functions {s m (z)} is defined by Let z ∈ σ (GA| T ) and 0 < α < 2 cos φ/ GA .It follows from the recursive form zs m+1 (z where an upper bound of β is given by (3.17).
The great attraction of the Newton-Raphson method is the generally quadratic nature of the convergence.Using the above facts in conjunction with Lemma 2.3, we see that a sequence {s m ( Ã)} defined by Thus we have the following corollary.
We remark that Corollary 4.1 is an extension of [4,13,15].It covers iterative methods for A † M,N in [17].The Newton-Raphson procedure can be speeded up by the successive matrix squaring technique in [14] if two parallel processors are available.In fact, the sequence in (4.5) is mathematically equivalent to (4.7) There are two matrix multiplications each step both in (4.5) and (4.7).However, A m+1 and P m+1 in (4.7) can be calculated simultaneously.Two algorithms given by (3.8) and (4.5) are also valid in the case when the spectrum of Ã is contained in the left half-plane with slight modification.
Moreover, all results in the previous two sections are valid without the restriction on σ (GA) if G is substituted by another matrix.This is stated as the following corollary.
As a matter of fact (4.9) is a direct result of [4,Lemma 3.4].
We remark a disadvantage of the choice G 0 of (4.8).In the case of computing A D with Ind(A) = k ≥ 3, G 0 = A k (A 2k+1 ) * A k , the condition number of G 0 A| T will be extremely large since cond(G 0 A| T ) = cond(A| T ) 4k+2 .An accurate numerical solution cannot be obtained if there is any round-off error in A.
It is remarked that the better accuracy of A d,w never be achieved and 1.7E-07 is the best error of is used as a stop criterion.This is because the condition number of GW AW | T is as large as 10 10 .If 2-step semi-iterative method of (3.25) is applied to compute A (2) T ,S , then {Y m } converges to A † after 54 iterations.However, the method fails to converge after 1500 iterations in other two cases because the segments [α, β] containing σ (GA| T ) are [−187970, 0.796] and [−355800, 0.9997], respectively, so that the rate of asymptotic convergence is too slow.
Example 5.2.Let A be 8 by 8 matrix with a complex spectrum given by   Example 5.3.Let r and c be a row vector and column vector, respectively, such that (5.2) A 10×16 complex Toeplitz matrix A is constructed by r and c.The stop criterion is the same as in Example 5.1.M and N are chosen positive definite diagonal matrix related to A, and W is a random matrix.The numbers of iterations by Newton's method and SIM method for A (2) T ,S are shown in Table 5.2.The data shows that Newton's method is much faster than that of SIM.

Received 10
May 2001 and in revised form 1 February 2002 , including special cases such as the weighted Moore-Penrose inverse A † M,N and the Drazin inverse A D .Numerical examples are given to illustrate our results.2000 Mathematics Subject Classification: 15A09, 65F20.

Corollary 4 . 2 .
Let A, T , S, and G be as in Lemma 2.1.Then Theorem 3.1 and Corollaries 3.2 and 4.1 are valid without any restriction on the spectrum of GA| T if G is substituted by G 0 = G(GAG) * G. (4.8) Proof.It suffices to show that

5 .
Examples.Three examples are given in this section to illustrate the computations of three types of A (2) T ,S .All calculations were performed on a PC with MATLAB.Example 5.1.Let A and W be 20 by 10 and 10 by 10 random matrices with entries on [−1, 1], respectively.We choose M and N as random symmetric and positive definite matrices of order 20 and 10, respectively.The stop criterion in (4.5) is A m−1 ∞ ≤ = 10 −10 .Three special cases A † , A † M,N , and A d,w are computed in this example.The choices of G, the number of iterations required and the norm of errors are listed in Table5.1.
a quadratic function of r on [r 1 ,r 2 ], which achieves its maximum at either r = r 1 or r = r 2 .So

Table 5 .
1. Newton-Raphson method for A † , A † M,N , and A d,w .