Estimation Of The Gross Output Vector Of Inter-Sectoral Balance Of Macroeconomic System

Solving the problem of the total least squares method on the basis of a normal biased system of equations

Problem (4) can be solved by the total least squares method. The problem is reduced to finding a vector

$\hat{X}=\underset{X\in {\mathbb{R}}^n}{\mathrm{m}\mathrm{i}\mathrm{n}}\frac{{‖\tilde{Y}-\left(I-\tilde{A}\right)X‖}_2^2}{1+{‖X‖}_2^2}$(5)

There are several approaches to minimizing (5). The first approach is based on the fact that the solution of problem (2) requires calculating the minimum singular number of the extended matrix $\left(\left(I-\tilde{A}\right),\tilde{Y}\right)$ and the corresponding right singular vector corresponding to this number.

The problem of finding a singular vector is a nonlinear vector problem. Its numerical solution is associated with significant difficulties associated with the stability of search algorithms, convergence issues, and high computational complexity.

The second approach is based on solving a biased normal system. In (Zhdanov, 1988) it is shown that if the condition

$\sigma ={\sigma }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(\left(I-\tilde{A}\right),\tilde{Y}\right)<{\sigma }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(I-\tilde{A}\right)$(6)

The solution to problem (4) can be obtained from a biased normal system of equations

$\left({\left(I-\tilde{A}\right)}^T\left(I-\tilde{A}\right)-{\sigma }_{}^{\mathrm{2}}I\right)X={\left(I-\tilde{A}\right)}^T\tilde{Y}$(7)

When solving system (7), only the scalar problem of finding the minimum singular number remains nonlinear ${\sigma }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(\left(I-\tilde{A}\right),\tilde{Y}\right).$

This task is always well-conditioned. The vector problem is linear. The system of linear algebraic equations (7) is always well-conditioned than the usual normal system. The condition number of the normal biased matrix is determined from the following relation:

$con{d}_{\mathrm{2}}\left(\left({\left(I-\tilde{A}\right)}^T\left(I-\tilde{A}\right)-{\sigma }_{}^{\mathrm{2}}I\right)\right)=\frac{{\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{2}}\left(I-\tilde{A}\right)-{\sigma }_{}^{\mathrm{2}}}{{\sigma }_{\mathrm{m}\mathrm{i}\mathrm{n}}^{\mathrm{2}}\left(I-\tilde{A}\right)-{\sigma }_{}^{\mathrm{2}}}.$(8)

Solving the problem of the total least-squares method using an augmented system of equations with complex coefficients

In (Zhdanov & Shamarov, 2000) it is proposed to use an augmented system equivalent to a biased normal system.

$C\stackrel{-}{X}=\stackrel{-}{Y},$ (9)

or

$\left(\begin{array}{ccc}{I}_m& {\mathrm{0}}_{m\times n}& k\left(I-\tilde{A}\right)\\ {\mathrm{0}}_{n\times m}& I_n& jk\sigma I_n\\ k{\left(I-\tilde{A}\right)}^T& jk\sigma I_n& \mathrm{0}{}_n\end{array}\right)\left(\begin{array}{c}kr\\ k{r}_{\sigma }\\ X\end{array}\right)=\left(\begin{array}{c}k\tilde{Y}\\ \mathrm{0}\\ \mathrm{0}\end{array}\right),$ (10)

wherе $k$ is arbitrary positive factor,

$j$ - is imaginary unit , $j=\sqrt{-\mathrm{1}}$.

The condition number of the matrix $\mathrm{}C\left(k\right)$ has the form

$con{d}_{\mathrm{2}}\left(C\left(k\right)\right)=\frac{\sqrt{\mathrm{1}+k^{\mathrm{2}}{\mu }_{\mathrm{m}\mathrm{a}\mathrm{x}}+k^{\mathrm{2}}{\sigma }_{}^{\mathrm{2}}}}{\sqrt{\mathrm{1}+k^{\mathrm{2}}{\mu }_{\mathrm{m}\mathrm{i}\mathrm{n}}+k^{\mathrm{2}}{\sigma }_{}^{\mathrm{2}}}}\left|\frac{\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathrm{1}}{\mathrm{3}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{2}}\frac{k^{\mathrm{2}}\left({\mu }_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\sigma }_{}^{\mathrm{2}}\right)}{{\left(\mathrm{1}+k^{\mathrm{2}}{\mu }_{\mathrm{m}\mathrm{a}\mathrm{x}}+k^{\mathrm{2}}{\sigma }_{}^{\mathrm{2}}\right)}^{\mathrm{3}/\mathrm{2}}}\right)\right)}{\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\pi }{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{3}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{2}}\frac{k^{\mathrm{2}}\left({\mu }_{\mathrm{m}\mathrm{i}\mathrm{n}}-{\sigma }_{}^{\mathrm{2}}\right)}{{\left(\mathrm{1}+k^{\mathrm{2}}{\mu }_{\mathrm{m}\mathrm{i}\mathrm{n}}+k^{\mathrm{2}}{\sigma }_{}^{\mathrm{2}}\right)}^{\mathrm{3}/\mathrm{2}}}\right)\right)}\right|,$ (11)

where ${\mu }_{\mathrm{m}\mathrm{a}\mathrm{x}},{\mu }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ are maximum and minimum eigenvalues of the matrix $\left(I-\tilde{A}\right){\left(I-\tilde{A}\right)}^T.$

The problem of finding the minimum value of the condition number can be considered as the problem of choosing the optimal factor $k$:

$\underset{k>\mathrm{0}}{\mathrm{i}\mathrm{n}\mathrm{f}}con{d}_{\mathrm{2}}\left(C\left(k\right)\right)$(12)

Problem (12) has no analytical solution, but can be solved by numerical methods. In practice, as an estimate, you can use

${\hat{k}}_{opt}=\frac{{\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}}^{}\left(I-\tilde{A}\right)+{\sigma }_{}^{}}{{\sigma }_{\mathrm{m}\mathrm{i}\mathrm{n}}^{}\left(I-\tilde{A}\right)+{\sigma }_{}^{}}\sqrt{\frac{\mathrm{2}}{{\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{2}}\left(I-\tilde{A}\right)+{\sigma }_{}^{\mathrm{2}}}}.$ (13)

Problem (11) can be solved, for example, using the LU decomposition.