Least Squares Solution of the Linear Operator Equation

Abstract

The least squares problems have wide applications in inverse Sturm–Liouville problem, particle physics and geology, inverse problems of vibration theory, control theory, digital image and signal processing. In this paper, we discuss the solution of the operator least squares problem. By extending the conjugate gradient least squares method, we propose an efficient matrix algorithm for solving the operator least squares problem. The matrix algorithm can find the solution of the problem within a finite number of iterations in the absence of round-off errors. Some numerical examples are given to illustrate the effectiveness of the matrix algorithm.

Appendix

1.1 The proof of Lemma 3.1

Step 1. Noting that the inner product is commutative, it is enough to prove three statements of Lemma 3.1 for $1\le u< v\le r.$ For $u=1$ and $v=2$, we get

$$\begin{aligned} \langle S(1),S(2)\rangle= & {} \langle S(1), S(1)-\delta (1)(\mathcal {F}^\mathrm{adj}(Q( 1))) \rangle \nonumber \\= & {} ||S(1)||^2-\delta (1)\langle \mathcal {F}(S(1)),Q( 1)\rangle \nonumber \\= & {} ||S(1)||^2-\delta (1)\langle Q( 1),Q( 1)\rangle =0, \end{aligned}$$

(26)

$$\begin{aligned} \langle Q(1),Q(2)\rangle= & {} \langle Q(1),\mathcal {F}(P(2))\rangle = \langle Q(1),\mathcal {F}(S(2)+\lambda (1)P(1))\rangle \nonumber \\= & {} \lambda (1)|| Q(1)||^2+\langle \mathcal {F}^\mathrm{adj}(Q(1)),S(2)\rangle \nonumber \\= & {} \lambda (1)|| Q(1)||^2-\frac{1}{\delta (1)}\langle \mathcal {F}^{adj}(R(2)-R(1)),S(2)\rangle \nonumber \\= & {} \lambda (1)|| Q(1)||^2-\frac{1}{\delta (1)}\langle S(2)-S(1),S(2)\rangle =0, \end{aligned}$$

(27)

and

$$\begin{aligned} = \langle P(1),S(2)\rangle =\langle S(1),S(2)\rangle =0. \end{aligned}$$

(28)

Hence, (I)–(III) hold for $u=1$ and $v=2$.

Step 2. In this step, for $u<w<r$, we assume that

$$\begin{aligned} \langle S(u),S(w)\rangle =0,~~~ \langle Q(u),Q(w)\rangle =0, ~~~ \langle P(u),S(w)\rangle =0. \end{aligned}$$

We can get

$$\begin{aligned} \langle S(u),S(w+1)\rangle= & {} \langle S(u),S(w)-\delta (w)(\mathcal {F}^{adj}(Q( w))\rangle =-\delta (w)\langle \mathcal {F}(S(u)),Q( w)\rangle \nonumber \\= & {} -\delta (w)\langle \mathcal {F}(P(u)-\lambda (u-1)P(u-1)),Q( w)\rangle \nonumber \\= & {} -\delta (w)\langle Q(u)-\lambda (u-1)Q(u-1),Q( w)\rangle =0, \end{aligned}$$

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$$\begin{aligned} \langle Q(u),Q(w+1)\rangle= & {} \langle Q(u),\mathcal {F}(P(w+1))\rangle =\langle Q(u),\mathcal {F}(S(w+1)+\lambda (w)P(w))\rangle \nonumber \\= & {} \langle Q(u),\mathcal {F}(S(w+1))+\lambda (w)Q(w)\rangle \nonumber \\= & {} -\frac{1}{\delta (u)}\langle R(u+1)-R(u),\mathcal {F}(S(w+1))\rangle \nonumber \\= & {} -\frac{1}{\delta (u)}\langle R(u+1)-R(u),\mathcal {F}(S(w+1))\rangle \nonumber \\= & {} -\frac{1}{\delta (u)}\langle \mathcal {F}^{adj}(R(u+1))-\mathcal {F}^{adj}(R(u)),S(w+1)\rangle \nonumber \\= & {} -\frac{1}{\delta (u)}\langle S(u+1)-S(u),S(w+1)\rangle =0, \end{aligned}$$

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and

$$\begin{aligned} \langle P(u),S(w+1)\rangle= & {} \langle P(u),S(w)-\delta (w)(\mathcal {F}^{adj}(Q(w))\rangle \nonumber \\= & {} -\delta (w)\langle \mathcal {F}(P(u)), Q(w)\rangle \nonumber \\= & {} -\delta (w)\langle Q(u), Q(w)\rangle =0. \end{aligned}$$

(31)

Furthermore, for $u=w$, we can write

$$\begin{aligned} \langle S(w),S(w+1)\rangle= & {} \langle S(w),S(w)-\delta (w)(\mathcal {F}^{adj}(Q( w))\rangle \nonumber \\= & {} ||S(w)||^2-\delta (w) \langle S(w),\mathcal {F}^{adj}(Q( w))\rangle \nonumber \\= & {} ||S(w)||^2-\delta (w) \langle \mathcal {F}( P(w)-\lambda (w-1)P(w-1)), Q( w)\rangle \nonumber \\= & {} ||S(w)||^2\!-\!\delta (w) \langle Q(w)\!-\!\lambda (w\!-\!1)Q(w\!-\!1), Q(w)\rangle \!=\!0,\quad \end{aligned}$$

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$$\begin{aligned} \langle Q(w),Q(w+1)\rangle= & {} \langle Q(w),\mathcal {F}(P(w+1))\rangle \nonumber \\= & {} \langle Q(w),\mathcal {F}(S(w+1)+\lambda (w)P(w))\rangle \nonumber \\= & {} \langle Q(w),\mathcal {F}(S(w+1))+\lambda (w)Q(w)\rangle \nonumber \\= & {} \lambda (w)||Q(w)||^2-\frac{1}{\delta (w)} \langle R(w+1)-R(w),\mathcal {F}(S(w+1))\rangle \nonumber \\= & {} \lambda (w)||Q(w)||^2-\frac{1}{\delta (w)} \langle \mathcal {F}^{adj}(R(w+1))\nonumber \\&-\mathcal {F}^\mathrm{adj}(R(w)),S(w+1)\rangle \nonumber \\= & {} \lambda (w)||Q(w)||^2-\frac{1}{\delta (w)} \langle S(w+1)-S(w),S(w+1)\rangle \nonumber \\= & {} \lambda (w)||Q(w)||^2-\frac{1}{\delta (w)}||S(w+1)||^2=0, \end{aligned}$$

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and

$$\begin{aligned} \langle P(w),S(w+1)\rangle= & {} \langle P(w),S(w)-\delta (w)(\mathcal {F}^{adj}(Q(w))\rangle \nonumber \\= & {} \langle S(w)+\lambda (w-1) S(w-1) +\lambda (w-1)\lambda (w-2) S(w-2)\nonumber \\&+cdots+\lambda (w-1)/cdots\lambda (1)S(1),S(w)\rangle \nonumber \\&-\!\delta (w)\langle \mathcal {F}(P(w)),Q(w)\rangle \!=\!||S(w)||^2\!-\delta (w)\langle Q(w),Q(w)\rangle \!=\!0.\nonumber \\ \end{aligned}$$

(34)

By considering Steps 1 and 2, three statements of Lemma 3.1 hold by the principle of induction.