Let I denote the \(n_{1}\times n_{1}\) unit matrix and let \(A\in \mathbb{C} _{r}^{n_{1}\times n_{2}}\) and the nonzero eigenvalues of \(A^{\ast }A\) be denoted by (1.4). We define the following matrix-valued functions consisting of matrix multiplications and additions:
$$\begin{aligned}& \varOmega (T_{m}) =T_{m}+T_{m}^{2}, \end{aligned}$$
(2.1)
$$\begin{aligned}& \varPsi (T_{m}) =I+T_{m}^{2}, \end{aligned}$$
(2.2)
$$\begin{aligned}& \varGamma (T_{m}) =T_{m}^{4}, \end{aligned}$$
(2.3)
where \(T_{m}= ( I-AV_{m} ) \). We propose the following:
Predictor-corrector iterative method (PCIM).
$$\begin{aligned}& In :V_{0}=\alpha A^{\ast } \\& G :T_{m}=I-AV_{m},\qquad \varPhi (T_{m})= \varPsi (T_{m})\varOmega (T_{m}) \\& P :V_{m+\frac{1}{2}}=V_{m} \bigl( I+\varPhi (T_{m}) \bigr) ,\qquad T _{m+\frac{1}{2}}=I-AV_{m+\frac{1}{2}} \\& C :V_{m+1}=V_{m+\frac{1}{2}} \bigl( I+\varPhi (T_{m+\frac{1}{2}}) \bigl( I+\varGamma (T_{m+\frac{1}{2}}) \bigr) \bigr) \\& \quad \text{for }m =0,1,2,\ldots. \end{aligned}$$
(2.4)
Here, In is the initial step, G is the generator, P is the predictor at the fractional step \(m+\frac{1}{2}\), and C is the corrector for the approximate Moore–Penrose inverse \(V_{m+1}\) of A at the \(m+1\)th iteration in (2.4). Let \(R_{m}=P_{R ( A ) }-AV_{m}\) be the residual at the mth iteration and consider the following form of (2.4) which requires \(P_{R ( A ) }\):
$$\begin{aligned}& In :V_{0}=\alpha A^{\ast }, \\& G :R_{m}=P_{R ( A ) }-AV_{m},\qquad \varPhi(R_{m})=\varPsi (R _{m})\varOmega (R_{m}), \\& P :V_{m+\frac{1}{2}}=V_{m} \bigl( P_{R(A)}+\varPhi (R_{m}) \bigr) ,\qquad R_{m+\frac{1}{2}}=P_{R ( A ) }-AV_{m+\frac{1}{2}}, \\& C :V_{m+1}=V_{m+\frac{1}{2}} \bigl( P_{R(A)}+\varPhi (R_{m+\frac{1}{2}}) \bigl( I+\varGamma (R_{m+\frac{1}{2}}) \bigr) \bigr) \\& \quad \text{for }m =0,1,2,\ldots \end{aligned}$$
(2.5)
Lemma 2.1
Let
\(A\in \mathbb{C} _{r}^{n_{1}\times n_{2}}\), if in the initial step
In
of (2.5
α
satisfies (1.3), then for the sequence
\(\{ V_{m+1} \} \)
obtained at
\(m+1\)th iteration by PCIM (2.5) the following hold true for every
\(m=0,1,2,\ldots\) :
$$\begin{aligned}& \mathrm{(i)}\quad V_{m+1}P_{R ( A ) } =V_{m+1},\qquad \mathrm{(ii)}\quad P_{R ( A^{\ast } ) }V_{m+1}=V_{m+1}, \end{aligned}$$
(2.6)
$$\begin{aligned}& \mathrm{(iii)}\quad ( AV_{m+1} ) ^{\ast } = AV_{m+1},\qquad \mathrm{(iv)}\quad ( V_{m+1}A ) ^{\ast }=V_{m+1}A. \end{aligned}$$
(2.7)
Proof
The proof follows from PCIM (2.5) using mathematical induction. We give the proof of (i) and (iii) and the proof of (ii) and (iv) can be given analogously. By \(( III ) \) of Theorem 1 [7], \(AA^{\dagger }=P_{R ( A ) }\) and \(A^{\dagger }A=P_{R ( A^{\ast } ) }\), and on the basis of Moore–Penrose equations (1.1), the projection \(P_{R ( A ) }\) satisfies the following:
$$ P_{R ( A ) }^{2}=P_{R ( A ) },\qquad P_{R (A )}^{\ast }=P_{R ( A ) },\qquad P_{R (A ) }A=A;\qquad A^{\ast }P_{R ( A ) }=A^{\ast }. $$
(2.8)
(i) For \(m=0\), in the initial step In of PCIM (2.5) we have \(V_{0}=\alpha A^{\ast }\) implying \(V_{0}P_{R ( A ) }= \alpha A^{\ast }P_{R ( A ) }=\alpha A^{\ast }=V_{0}\). Assume that \(V_{m}P_{R ( A ) }=V_{m}\) holds true for m, then using (2.8) for the predictor step P of (2.5) we obtain
$$\begin{aligned} &\begin{aligned}[b] V_{m+\frac{1}{2}}P_{R ( A ) } ={}&V_{m} \bigl( P_{R(A)}+ \varPhi (R_{m}) \bigr) P_{R ( A ) } \\ ={}&V_{m} \bigl( P_{R ( A ) }+ ( P_{R ( A ) }-AV_{m} ) + ( P_{R ( A ) }-AV_{m} ) ^{2}+ ( P_{R ( A ) }-AV_{m} ) ^{3} \\ &{}+ ( P_{R ( A ) }-AV_{m} ) ^{4} \bigr) P_{R ( A ) }=V_{m+\frac{1}{2}}, \end{aligned} \end{aligned}$$
(2.9)
$$\begin{aligned} &\begin{aligned}[b] R_{m+\frac{1}{2}} &=P_{R(A)}-AV_{m+\frac{1}{2}}=P_{R(A)}-AV_{m} \bigl( P_{R(A)}+\varPhi (R_{m}) \bigr) \\ &=P_{R(A)}-AV_{m} \bigl(P_{R(A)}+R_{m}+R_{m}^{2}+R_{m}^{3}+R_{m}^{4} \bigr)=R _{m}^{5}. \end{aligned} \end{aligned}$$
(2.10)
Also in the corrector step C of (2.5) using (2.10) we get
$$\begin{aligned}& \varGamma (R_{m+\frac{1}{2}}) =R_{m+\frac{1}{2}}^{4}=R_{m}^{20}, \end{aligned}$$
(2.11)
$$\begin{aligned}& \varPhi (R_{m+\frac{1}{2}}) =\varPsi (R_{m+\frac{1}{2}})\varOmega (R_{m+ \frac{1}{2}})= \bigl( I+R_{m}^{10} \bigr) \bigl( R_{m}^{5}+R_{m} ^{10} \bigr) . \end{aligned}$$
(2.12)
Using (2.8) and from the assumption \(V_{m}P_{R ( A ) }=V_{m}\), equations (2.11) and (2.12) imply
$$\begin{aligned}& \varGamma (R_{m+\frac{1}{2}})P_{R ( A ) }= ( P_{R ( A ) }-AV_{m} ) ^{20}P_{R ( A ) }=\varGamma (R_{m+ \frac{1}{2}}), \end{aligned}$$
(2.13)
$$\begin{aligned}& \varPhi (R_{m+\frac{1}{2}})P_{R ( A ) }= \bigl( I+R_{m}^{10} \bigr) \bigl( R_{m}^{5}+R_{m}^{10} \bigr) P_{R ( A ) }=\varPhi (R_{m+\frac{1}{2}}). \end{aligned}$$
(2.14)
Using (2.8)–(2.14) in the corrector step C of (2.5), we obtain
$$\begin{aligned} V_{m+1}P_{R ( A ) } =&V_{m+\frac{1}{2}} \bigl( P_{R(A)}+ \varPhi (R_{m+\frac{1}{2}}) \bigl( I+\varGamma (R_{m+\frac{1}{2}}) \bigr) \bigr) P_{R ( A ) } \\ =&V_{m+\frac{1}{2}} \bigl( P_{R(A)}+\varPhi (R_{m+\frac{1}{2}})P_{R ( A ) }+ \varPhi (R_{m+\frac{1}{2}})\varGamma (R_{m+\frac{1}{2}})P _{R ( A ) } \bigr) \\ =&V_{m+\frac{1}{2}} \bigl( P_{R(A)}+\varPhi (R_{m+\frac{1}{2}}) \bigl( I+ \varGamma (R_{m+\frac{1}{2}}) \bigr) \bigr) =V_{m+1}. \end{aligned}$$
(2.15)
(iii) For \(m=0\), in the initial step In of PCIM (2.5) we have \(V_{0}=\alpha A^{\ast }\) implying \(( AV_{0} ) ^{\ast }= ( A\alpha A^{\ast } ) ^{\ast }=\alpha AA^{\ast }=AV_{0}\). Assume that \(( AV_{m} ) ^{\ast }=AV_{m}\) holds true, then from (2.8) in the predictor step P of (2.5) gives
$$\begin{aligned}& R_{m}^{\ast } = ( P_{R(A)}-AV_{m} ) ^{\ast }=R_{m}, \end{aligned}$$
(2.16)
$$\begin{aligned}& \varPhi ^{\ast }(R_{m}) = \bigl( \bigl( I+R_{m}^{10} \bigr) \bigl( R _{m}^{5}+R_{m}^{10} \bigr) \bigr) ^{\ast }=\varPhi (R_{m}), \end{aligned}$$
(2.17)
$$\begin{aligned}& ( AV_{m+\frac{1}{2}} ) ^{\ast } = \bigl( P_{R(A)}^{ \ast }+ \varPhi ^{\ast }(R_{m}) \bigr) ( AV_{m} ) ^{\ast }=AV _{m+\frac{1}{2}}. \end{aligned}$$
(2.18)
From the assumption and also using (2.8), (2.10)–(2.12), and (2.16) gives
$$ R_{m+\frac{1}{2}}^{\ast }=R_{m+\frac{1}{2}},\qquad \varGamma ^{\ast }(R_{m+ \frac{1}{2}})=\varGamma (R_{m+\frac{1}{2}}),\qquad \varPhi ^{\ast }(R_{m+ \frac{1}{2}})=\varPhi (R_{m+\frac{1}{2}}). $$
(2.19)
Using (2.8), (2.18), and (2.19) in the corrector step C of (2.5) yields
$$\begin{aligned} ( AV_{m+1} ) ^{\ast } =& \bigl( P_{R(A)}+\varPhi (R_{m+ \frac{1}{2}}) \bigl( I+\varGamma (R_{m+\frac{1}{2}}) \bigr) \bigr) ^{\ast } ( AV_{m+\frac{1}{2}} ) ^{\ast } \\ =&AV_{m+1}. \end{aligned}$$
(2.20)
□
Remark 2.2
The validity of (2.6) and (2.7) for PCIM (2.5) implies that PCIM (2.4) is derived from PCIM (2.5). The computational significance of method (2.5) is limited by the need for knowledge of \(P_{R(A)}\).Therefore, the PCIM given in (2.4) is more effective in computational aspects and is preferable for \(n_{1}\leq n_{2}\). If \(n_{1}>n_{2}\), then the dual version of PCIM (2.4) can be used.
Theorem 2.3
Let
\(A\in \mathbb{C} _{r}^{n_{1}\times n_{2}}\), if in the initial step
In
of PCIM (2.4) α
satisfies (1.3), then the sequence
\(\{ V_{m+1} \} \)
obtained by the proposed PCIM (2.4) converges to the Moore–Penrose inverse
\(A^{\dagger }\)
uniformly with
\(p=45\)
order of convergence and with asymptotic convergence factor [18] \(ACF=\frac{10}{\ln 45}\approx 2.627\). Furthermore, the following error estimate is valid:
$$ \bigl\Vert A^{\dagger }-V_{m+1} \bigr\Vert _{2} \leq \frac{\alpha \Vert R_{0} \Vert _{2}^{45^{m+1}} \Vert A^{\ast } \Vert _{2}}{1- \Vert R_{0} \Vert _{2}}, $$
(2.21)
where
\(R_{0}=P_{R ( A ) }-\alpha AA^{\ast }\).
Proof
To show the uniform convergence of \(\{ V_{m+1} \} \) obtained by PCIM (2.4) to \(A^{\dagger }\), we use the approach analogous to the proof of Theorem 3 of [7]. Let us define the residual in the dual version at initial approximation by
$$ \overline{T}_{0}=I-V_{0}A=I-\alpha A^{\ast }A. $$
(2.22)
Also, for any integer \(s>0\),
$$ T_{0}^{s}A=A\overline{T}_{0}^{s} $$
(2.23)
and for PCIM (2.4), for its dual version and from Lemma 2.1 using (2.10)–(2.12) for PCIM (2.5), we obtain the residual error at the \((m+1)\)th iteration as
$$\begin{aligned}& T_{m+1} =I-AV_{m+1}=T_{m}^{45}, \quad m=0,1,\ldots, \end{aligned}$$
(2.24)
$$\begin{aligned}& \overline{T}_{m+1} =I-V_{m+1}A= \overline{T}_{m}^{45},\quad m=0,1,\ldots, \end{aligned}$$
(2.25)
$$\begin{aligned}& R_{m+1} =P_{R ( A ) }-AV_{m+1}=R_{m}^{45}, \quad m=0,1,\ldots, \end{aligned}$$
(2.26)
respectively. It follows from (2.24)–(2.26) that, for each m,
$$\begin{aligned}& AV_{m+1} =I-T_{0}^{45^{m+1}},\qquad V_{m+1}A=I-\overline{T}_{0} ^{45^{m+1}}, \end{aligned}$$
(2.27)
$$\begin{aligned}& V_{m+1}AV_{m+1} =V_{m+1}- \overline{T}_{0}^{45^{m+1}}V_{m+1}. \end{aligned}$$
(2.28)
From Lemma 2.1, equations (2.6), (2.7) and on the basis of Corollary 4 to Theorem 2 and Theorem 1 of [7] the passage to the limit in (2.27) and (2.28) imply the uniform convergence
$$ AV_{m+1}\rightarrow P_{R ( A ) },\qquad V_{m+1}A\rightarrow P_{R ( A^{\ast } ) },\qquad V_{m+1}AV_{m+1}\rightarrow A ^{\dagger }. $$
(2.29)
Therefore, the sequence \(\{ V_{m+1} \} \) converges uniformly to \(A^{\dagger }\) with order \(p=45\). By denoting
$$ \widehat{V}_{m+\frac{1}{2}}=V_{m+\frac{1}{2}} \bigl( I+T_{m}^{5}+T _{m}^{10}+T_{m}^{15}+T_{m}^{20}+T_{m}^{25}+T_{m}^{30}+T_{m}^{35} \bigr), $$
(2.30)
the corrector step C of (2.4) can be rewritten as
$$ V_{m+1}=V_{m+\frac{1}{2}}+\widehat{V}_{m+\frac{1}{2}}T_{m}^{5}. $$
(2.31)
We denote the error of PCIM (2.4) at the corrector step C by \(E_{m+1}=A^{\dagger }-V_{m+1}\). Next, by using that \(A^{\dagger }AA ^{\dagger }=A^{\dagger }\) and from Lemma 2.1 equations (2.6) and (2.8), we obtain
$$\begin{aligned} E_{m+1}-E_{m+1}R_{m+1} =&A^{\dagger }-V_{m+1}- \bigl( A^{\dagger }-V _{m+1} \bigr) ( P_{R ( A ) }-AV_{m+1} ) \\ =&V_{m+1} \bigl( AA^{\dagger }-AV_{m+1} \bigr) =V_{m+1}R_{m+1}. \end{aligned}$$
(2.32)
Using (1.3) and on the basis of Theorem 2 of [6], we have \(\rho ( R_{0} ) <1\). Since \(R_{0}\) is Hermitian, this gives \(\rho ( R_{0} ) = \Vert R _{0} \Vert _{2}<1\) and using (2.26) and norm properties
$$ \Vert R_{m+1} \Vert _{2}\leq \Vert R_{m} \Vert _{2}^{45}\leq \Vert R_{0} \Vert _{2}^{45^{m+1}}< 1. $$
(2.33)
From (2.32) using second norm gives
$$ \Vert E_{m+1} \Vert _{2}\leq \frac{ \Vert V_{m+1}R _{m+1} \Vert _{2}}{1- \Vert R_{m+1} \Vert _{2}} $$
(2.34)
(see [7]). By using (2.26), (2.30), and (2.31), it follows that
$$\begin{aligned} V_{m+1}R_{m+1} =&V_{m+1}R_{m}^{40}R_{m}^{5} \\ =& \bigl\{ V_{m+1} \bigl( P_{R ( A ) }+R_{m}^{5}+R_{m} ^{10}+R_{m}^{15}+R_{m}^{20}+R_{m}^{25}+R_{m}^{30}+R_{m}^{35}+R_{m} ^{40} \bigr) \\ &{}- V_{m+1} \bigl( P_{R ( A ) }+R_{m}^{5}+R_{m}^{10}+R _{m}^{15}+R_{m}^{20}+R_{m}^{25}+R_{m}^{30}+R_{m}^{35} \bigr) \bigr\} R_{m}^{5} \\ =& \bigl\{ V_{m+1} \bigl( P_{R ( A ) }+R_{m}^{5}+R_{m} ^{10}+R_{m}^{15}+R_{m}^{20}+R_{m}^{25}+R_{m}^{30}+R_{m}^{35}+R_{m} ^{40} \bigr) \\ &{}-V_{m+\frac{1}{2}} \bigl( P_{R ( A ) }+R_{m}^{5}+R_{m} ^{10}+R_{m}^{15}+R_{m}^{20}+R_{m}^{25}+R_{m}^{30}+R_{m}^{35} \bigr) \\ &{}-\widehat{V}_{m+\frac{1}{2}}T_{m}^{5} \bigl( P_{R ( A ) }+R_{m}^{5}+R_{m}^{10}+R_{m}^{15}+R_{m}^{20}+R_{m}^{25}+R _{m}^{30}+R_{m}^{35} \bigr) \bigr\} R_{m}^{5} \\ =& ( V_{m+1}-\widehat{V}_{m+\frac{1}{2}} ) \bigl( P_{R ( A ) }+R_{m}^{5}+R_{m}^{10}+R_{m}^{15}+R_{m}^{20} \\ &{} +R_{m}^{25}+R_{m}^{30}+R_{m}^{35}+R_{m}^{40} \bigr) R_{m} ^{5}. \end{aligned}$$
(2.35)
From equation (2.35) and using \(\Vert P_{R ( A ) } \Vert _{2}=1\) and second norm, we get
$$\begin{aligned} \Vert V_{m+1}R_{m+1} \Vert _{2} \leq & \Vert R_{m} \Vert _{2} ^{5} \bigl( 1+ \Vert R_{m} \Vert _{2}^{5}+ \Vert R _{m} \Vert _{2}^{10}+ \Vert R_{m} \Vert _{2}^{15}+ \Vert R_{m} \Vert _{2}^{20}+ \Vert R_{m} \Vert _{2}^{25} \\ &{} + \Vert R_{m} \Vert _{2}^{30}+ \Vert R_{m} \Vert _{2}^{35}+ \Vert R_{m} \Vert _{2}^{40} \bigr) \Vert V _{m+1}-\widehat{V}_{m+\frac{1}{2}} \Vert _{2}. \end{aligned}$$
(2.36)
From (2.30) and (2.31) and using that \(V_{m+\frac{1}{2}}P_{R(A)}^{s}=V_{m+\frac{1}{2}}\) is valid for every positive integer s and also using second norm, it follows that
$$\begin{aligned} \Vert V_{m+1}-\widehat{V}_{m+\frac{1}{2}} \Vert _{2} =& \bigl\Vert V_{m+\frac{1}{2}}+\widehat{V}_{m+\frac{1}{2}} \bigl( T _{m}^{5}-I \bigr) \bigr\Vert _{2} \\ =& \bigl\Vert V_{m+\frac{1}{2}}T_{m}^{40} \bigr\Vert _{2}\leq \Vert V _{m+\frac{1}{2}}R_{m} \Vert _{2} \Vert R_{m} \Vert _{2}^{39}. \end{aligned}$$
(2.37)
Also, in view of (2.33),
$$\begin{aligned} 1- \Vert R_{m+1} \Vert _{2} \geq &1- \Vert R_{m} \Vert _{2}^{45}= \bigl( 1- \Vert R_{m} \Vert _{2}^{5} \bigr) \bigl( 1+ \Vert R_{m} \Vert _{2}^{5}+ \Vert R_{m} \Vert _{2}^{10}+ \Vert R_{m} \Vert _{2}^{15} \\ &{} + \Vert R_{m} \Vert _{2}^{20}+ \Vert R_{m} \Vert _{2}^{25}+ \Vert R_{m} \Vert _{2}^{30}+ \Vert R_{m} \Vert _{2}^{35}+ \Vert R_{m} \Vert _{2}^{40}\bigr) . \end{aligned}$$
(2.38)
From the predictor step P of (2.4) and using \(\Vert P _{R ( A ) } \Vert _{2}=1\) and second norm, it follows that
$$ \Vert V_{m+\frac{1}{2}}R_{m} \Vert _{2}\leq \Vert V _{m}R_{m} \Vert _{2} \bigl( 1+ \Vert R_{m} \Vert _{2}+ \Vert R_{m} \Vert _{2}^{2}+ \Vert R_{m} \Vert _{2}^{3}+ \Vert R_{m} \Vert _{2}^{4} \bigr) . $$
(2.39)
Also
$$ 1- \Vert R_{m} \Vert _{2} ^{5}= \bigl( 1- \Vert R_{m} \Vert _{2} \bigr) \bigl( 1+ \Vert R_{m} \Vert _{2}+ \Vert R _{m} \Vert _{2}^{2}+ \Vert R_{m} \Vert _{2}^{3}+ \Vert R_{m} \Vert _{2}^{4} \bigr) . $$
(2.40)
Using (2.34), (2.36)–(2.40) and by recursion, we obtain the following inequalities yielding (2.21):
$$\begin{aligned} \bigl\Vert A^{\dagger }-V_{m+1} \bigr\Vert _{2} \leq &\frac{ \Vert V _{m+1}R_{m+1} \Vert _{2}}{1- \Vert R_{m+1} \Vert _{2}}\leq \frac{ \Vert R_{m} \Vert _{2}^{44} \Vert V _{m+\frac{1}{2}}R_{m} \Vert _{2}}{1- \Vert R_{m} \Vert _{2}^{5}} \leq \frac{ \Vert R_{m} \Vert _{2}^{44} \Vert V_{m}R _{m} \Vert _{2}}{1- \Vert R_{m} \Vert _{2}} \\ \leq& \frac{ \Vert R_{0} \Vert _{2}^{45^{m+1}} \Vert V_{0} \Vert _{2}}{1- \Vert R_{0} \Vert _{2}} =\frac{\alpha \Vert R_{0} \Vert _{2}^{45^{m+1}} \Vert A ^{\ast } \Vert _{2}}{1- \Vert R_{0} \Vert _{2}}. \end{aligned}$$
(2.41)
The proposed PCIM (2.4) requires 5 \(Mms\) for the evaluation of the predictor step P and 5 \(Mms\) for the corrector step C. Therefore, total 10 \(Mms\) are needed giving the asymptotic convergence factor \(ACF\) [18] as \(\frac{10}{\ln 45} \approx 2.627\). □