On the numerical solution of the general kinetic “K-angle” reaction system

Abstract

In a recent paper by Tobias and Tasi in this journal the authors discuss the “K-angle” kinetic problem and focus on its analytic solution for $K\le 5$. The authors correctly mention that the case $K>5$ requires numerical treatment in the part of the procedure where the eigenvalues of the characteristic polynomial are determined by solving an algebraic equation of degree greater than or equal to five. In this work we focus on this algorithmic part by suggesting a suitable and effective numerical procedure. For the sake of simplifying the numerical procedure and to formulate the whole algorithm in an easy programmable way we reformulate the “K-angle” problem by introducing double indices for the reactions between the components. This notation allows to greatly simplify the formulation of the problem and the description of the whole algorithmic procedure and facilitates the creation of appropriate programming tools. We present an example of such a tool, namely a software module named K-angle within the programming environment of CAS Mathematica possessing certain visualization properties.

Appendix

1.1 Problems related to the study of the characteristic polynomial of nth degree for the computation of the eigenvalues

We consider the characteristic polynomial $D(\lambda )$ of the $n \times n$ matrix A

$$\begin{aligned} D(\lambda )=\left| A-\lambda I\right| =(-1)^n\left( \lambda ^n+p_1\lambda ^{n-1}+\cdots +p_{n-1}\lambda +p_n\right) \end{aligned}$$

(4)

with simple real or complex zeros: $\lambda _1, \lambda _2 ,\ldots ,\lambda _n$.

One of the most effective methods for the simultaneous determination of the roots of a polynomial, the origin of which is attributed to Weierstrass [18], is the one proposed by Kerner and, independently, by Durand and Dochev. The method known as the Weierstrass method, the Durand–Kerner method, or the Weierstrass–Dochev method.

The simultaneous computation of the zeros of (4) by the Weierstrass method follows the scheme:

$$\begin{aligned} \lambda _i^{k+1}=\lambda _i^k- \frac{{\displaystyle D\left( \lambda _i^k\right) }}{{\displaystyle \prod ^n_{j\ne i}\left( \lambda _i^k-\lambda _j^k\right) }},\, \quad i=1,\ldots ,n; \, \, k=0,1,2,\ldots . \end{aligned}$$

(5)

The convergence rate of the scheme (5) is given in the following:

Theorem

[11] Let $0<q<1$, $d={\displaystyle \min _{i\ne j}}|\lambda _i-\lambda _j|$ and $0<c\le {d}/{1+\alpha n}$, where $\alpha = 1.7632283\ldots$ is determined by the equality $\alpha = e^{{1}/{\alpha }}$. If the initial approximations $\{\lambda _i^0\}, \, i=1,2,\ldots ,n$ of the roots $\{\lambda _i\}; \, i=1,2,\ldots ,n$ of the equation $D(\lambda )=0$ satisfy the inequalities $|\lambda _i^0-\lambda _i|\le cq,\, i=1,2,\ldots ,n$, then for the approximations given by (5) the inequalities

$$\begin{aligned} |\lambda _i^k-\lambda _i|\le cq^{2^k}, \quad i=1,2,\ldots ,n. \end{aligned}$$

(6)

hold true for all $k=1,2,\ldots .$

The method (5) has been modified for multiple roots, see [5, 7, 16].

Remark

Let $D(\lambda )$ be the polynomial with real or complex multiple zeros $\lambda _1,\ldots$, $\lambda _m$ with the multiplicities $s_i,\, i=1,\ldots ,m$, where ${\displaystyle \sum _{i=1}^m} s_i=n$, i.e.

$$\begin{aligned} D(\lambda )=\prod _{j=1}^m(\lambda -\lambda _j)^{s_j}. \end{aligned}$$

In this case we can use the following procedure [10]:

$$\begin{aligned} \lambda _i^{k+1}=\lambda _i^k-\frac{s_i}{H(\lambda _i^k)-{\displaystyle \sum _{l\ne i}^m}\frac{s_l}{\lambda _i^k-\lambda _l^k-\Delta _l^{R,k}}},\quad i=1,\ldots ,m;\, k=0,1,2,\ldots . \end{aligned}$$

where

$$\begin{aligned} H(\lambda _i^k)= & {} \frac{D^{'}(\lambda _i^k)}{D(\lambda _i^k)} \\ \Delta _t^{R,k}= & {} -s_i\left( H(\lambda _i^k)-{\displaystyle \sum _{q\ne t}^m}\frac{s_q}{\lambda _t^k-\lambda _q^k-\Delta _q^{R-1,k}}\right) ^{-1},\, t=1,2,\ldots ,m \\ \Delta _t^{0,k}= & {} 0,\, t=1,2,\ldots ,m;\, k=0,1,2,\ldots . \end{aligned}$$

The following theorem holds

Theorem

[10] Let $0<q<1$, $d={\displaystyle \min _{i\ne j}}|\lambda _i-\lambda _j|$ and $c>0$ is a number such that

$$\begin{aligned}&d>c(2+3qn/s),\\&c^2(n-s^{'})\left( (d-c)(d-2c-sqn/s)\left( 1-\frac{(n/s+1)(n/s^{'}-1)c^2q^2}{(d-c)(d-2c-sqn/s)}\right) \right) ^{-1}<1,\\&s={\displaystyle \max _{1\le j\le m}} s_j;\, \, s^{'}={\displaystyle \min _{1\le j\le m}} s_j. \end{aligned}$$

If the initial approximations $\{\lambda _i^0\}, \, i=1,2,\ldots ,m$ of the roots $\{\lambda _i\}; \, i=1,2,\ldots ,m$ satisfy the inequalities $|\lambda _i^0-\lambda _i|\le cq,\, i=1,2,\ldots ,m$, then the estimate

$$\begin{aligned} |\lambda _i^k-\lambda _i|\le cq^{(2R+3)^k}, \, i=1,2,\ldots ,m; \, k=0,1,2,\ldots \end{aligned}$$

hold true.

1.2 The Le Verrier method in terms of the Weierstrass procedure

We next give an alternative formulation of the Le Verrier-Fadeev method in the terms of the Weierstrass root-finding method. Some computational aspects are discussed. From (4) we have

$$\begin{aligned} p_m=(-1)^m{\displaystyle \sum _{1\le i_1<i_2<\cdots <i_m\le n}}\lambda _{i_1}\lambda _{i_2} \cdots \lambda _{i_m} \end{aligned}$$

The Newton-Girard formulae give the connection between the coefficients $p_r$, and the power sums:

$$\begin{aligned} S_r=\lambda _1^r+\cdots +\lambda _n^r \end{aligned}$$

$$\begin{aligned} S_r+S_{r-1}p_1+S_{r-2}p_2+\cdots +S_1p_{r-1}+rp_r=0,\, (r\le n). \end{aligned}$$

(7)

From (7) it follows

$$\begin{aligned}&p_1=-S_1, \nonumber \\&{\displaystyle p_2=-\frac{1}{2}(S_1p_1+S_2),} \nonumber \\&\ldots \nonumber \\&{\displaystyle p_r=-\frac{1}{r}(S_1p_{r-1}+S_2p_{r-2}+\cdots +S_{r-1}p_1+S_r).} \end{aligned}$$

(8)

Obviously, $S_r=\mathrm{Trace}\left( A^r\right)$ and from (8) we obtain the coefficients $p_r$.

The process is called Le Verrier method. In a number of cases the calculations are quite difficult.

An alternative formulation of Le Verrier’s method is the following:

Let $C(\lambda )$ be the companion matrix of $A-\lambda I$ such that

$$\begin{aligned} C(\lambda )\left( A-\lambda I\right) =D(\lambda )I. \end{aligned}$$

We form the matrix $C(\lambda )$ in the following way

$$\begin{aligned} C(\lambda )=C_0\lambda ^{n-1}+C_1\lambda ^{n-2}+\cdots +C_{n-2}\lambda +C_{n-1}, \end{aligned}$$

where $C_i,\, i=0,1,\ldots , n-1$ are $n \times n$ matrices, independent of $\lambda$. Then the matrices $C_j$ are immediately computed from the following system of Fadeev identities:

$$\begin{aligned}&-C_0=(-1)^nI, \nonumber \\&C_1-C_0A=(-1)^{n-1}p_1I, \nonumber \\&C_2-C_1A=(-1)^{n-1}p_2I, \nonumber \\&\ldots \nonumber \\&C_{n-1}-C_{n-2}A=(-1)^{n-1}p_{n-1}I, \nonumber \\&C_{n-1}A=(-1)^np_nI. \end{aligned}$$

(9)

Given two distinct approximations $\lambda _i^k,\, \lambda _i^{k+1},\, i=1,2,\ldots ,n$, obtained from the Weierstrass series (5).

We have [8]:

(10)

Using (9) and (10) we obtain [9]:

$$\begin{aligned}&C_1=(-1)^{n-1}\left[ A-{\displaystyle \sum _{i=1}^n}\lambda _i^{k+1}\right] I, \\&\mathrm{Trace}\left( C_1A\right) =(-1)^n2\left( {\displaystyle \sum ^n_{i=1}}\lambda _i^{k+1}{\displaystyle \sum ^n_{j\ne i}}\lambda _j^k- {\displaystyle \sum ^n_{l<s}}\lambda _l^k\lambda _s^k\right) , \nonumber \\&\ldots \nonumber \\&\mathrm{Trace}\left( C_{n-1}A\right) =n\left( {\displaystyle \sum ^n_{i=1}}\lambda _i^{k+1} {\displaystyle \prod ^n_{j\ne i}}\lambda _j^k-(n-1){\displaystyle \prod ^n_{j=1}}\lambda _j^k\right) , \end{aligned}$$

and consequently we find $\det |A|$, the companion matrix, and consequently we can determine the matrix $A^{-1}$.