**4.1. The argument principle of meromorphic function**

In the basis of algorithm of finding number of zeros and their approximations, which are in some areas *G* , is the statement that follows from the argument principle of meromorphic functions.

**Statement**. *Let the meromorphic function f*( )  *have in the region G m zeros* 1 2 , , , *<sup>m</sup> (with regard for their multiplicity) and no zeros on the boundary of region G , then the number m is determined in accordance with the principle of the argument* 

$$m = s\_0 = \frac{1}{2\pi i} \int\_{\Gamma} \frac{f'(\lambda)}{f(\lambda)} d\lambda \tag{23}$$

Numerical Algorithms of Finding the Branching Lines

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 297

(28)

The found eigenvalues can be refined, using them as initial approximations for Newton's

*l*

*l <sup>f</sup> <sup>l</sup>*

 

2 1

*l*

2 1

The argument principle (23) and formula of the argument principle type (24), (25) were repeatedly applied in solving various spectral problems (see, for example, [1, 7, 9] and references therein), but the peculiarity of the proposed algorithm is to compute the values of

and its derivatives basing on *LU*-decomposition of the matrix ( ) *<sup>n</sup>* **T**

( ) det ( ) ( ) ( ),

( ) det ( ) ( ) ( ) ( ) ( ) ( ) ,

**D LU** ( ) ( ) ( ),

**B MU LV** ( ) ( ) ( ) ( ) ( ),

**C =N U + M V +L W**

 

 

 *and* ( ) *wii* 

 *is the lower triangular matrix with single diagonal elements.*

 *in decompositions* 

 

 *for derivatives of determinant* det ( ) ( ) **D**

1 1,

1 1, 1 1, 1, ,

 

> 

*n n n n n*

*k k i i k i ik*

*k k i i k k j j i i k i ik k j jk i i ki j*

 *w uv v u* 

**<sup>D</sup>** (31)

 *v u* 

**<sup>D</sup>** (30)

*n n*

**4.2. Numerical procedure of calculating the derivatives (the first and the second)** 

*l <sup>f</sup> <sup>l</sup>*

*f* 

( ) , 0,1,2, ( )

2 2

*f f f ff* 

2 22 ( )( ) ( ) ( )( ) *l l*

( ) , 0,1,2, ( )

*l ll*

 

> 

(29)

.

 *of matrix* **D**( )

   *the* 

 *are differentiable functions with respect to* 

 

(32)

(33)

, (34)

 *are, respectively, the elements of the upper triangular* 

 *f* 

1

**Theorem**. *If the elements of square matrix* **D**( )

*f* 

 *and* **W**( ) 

*l l*

or for one of bilateral analogies of Newton's method [15], for example,

*l l*

22 21

 

*l l*

 

 

21 2 2

*f* 

method

function *f*( )

*parameter* 

*relations* 

*f* 

*matrix* **U**( ) *,* **V**( ) 

and **L**( ) 

*are true, where* ( ), ( ) *ii ii u v*

**for the matrix determinant** 

*, then for any* 

*and relations* 

$$\sum\_{j=1}^{m} \left(\mathcal{A}\_{j}\right)^{k} = \mathbf{s}\_{k'} \qquad k = 1, \ldots, m \tag{24}$$

*are true, where* 

$$s\_k = \frac{1}{2\pi i} \int\_{\Gamma} \mathcal{X} \frac{f'(\lambda)}{f(\lambda)} d\lambda, \quad k = 0, 1, \ldots \tag{25}$$

Thus, knowing , 1,2, , *<sup>k</sup> sk m* , from the system (24) we can find the zeros of functions *f*( ) that are in the region *G* .

By putting the interval , *t t c d* **[ ]** in the region *G* , such as a circle with center at <sup>0</sup> ( )2 *t tt c d <sup>r</sup>* **/** and radius ( )2 *t t t dc* **/** , and applying the above statement to the meromorphic function ( ) det ( ) *<sup>n</sup> f* **T** , you can find all the eigenvalues of problem (22), belonging to the given region *G* , i.e. to the given interval , *t t c d* **[ ]** . The integrals in (23) and (25) we can replace by some approximate quadrature formulas, such as rectangles at *N* points on , and since is a circle, then to calculate quantities , 0,1,2, *<sup>k</sup> s k* , we obtain the relation

$$s\_k = \frac{1}{N} \sum\_{j=1}^{N} (\lambda\_j)^k \rho\_t \exp\left(i \frac{2\pi j}{N}\right) \frac{f'(\lambda\_j)}{f(\lambda\_j)},\tag{26}$$

where 0 2 exp *<sup>t</sup> j t <sup>j</sup> r i <sup>N</sup>* . The system itself (24) we solve using Newton's method, by choosing the initial approximation on the border of the region *G* :

$$\mathcal{A}\_{j}^{(0)} = r\_{0\_{j}} + \rho\_{t} \exp\left(i\frac{2\pi j}{s\_{0}}\right), \quad j = 1, 2, \ldots, s\_{0}. \tag{27}$$

The found eigenvalues can be refined, using them as initial approximations for Newton's method

$$
\lambda\_{l+1} = \lambda\_l - \frac{f(\lambda\_l)}{f'(\lambda\_l)}, \quad l = 0, 1, 2, \dots \tag{28}
$$

or for one of bilateral analogies of Newton's method [15], for example,

296 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Statement**. *Let the meromorphic function f*( )

*determined in accordance with the principle of the argument* 

functions.

*and relations* 

*are true, where* 

<sup>0</sup> ( )2 *t tt c d <sup>r</sup>* 

the relation

where 0

that are in the region *G* .

meromorphic function ( ) det ( ) *<sup>n</sup> f*

By putting the interval ,

exp *<sup>t</sup> j t <sup>j</sup> r i <sup>N</sup>*

 

*f*( ) 

**4.1. The argument principle of meromorphic function** 

In the basis of algorithm of finding number of zeros and their approximations, which are in some areas *G* , is the statement that follows from the argument principle of meromorphic

*regard for their multiplicity) and no zeros on the boundary of region G , then the number m is* 

( ) , 1, ,

*sk m*

<sup>1</sup> ( ) , 0,1, . 2 ()

 

Thus, knowing , 1,2, , *<sup>k</sup> sk m* , from the system (24) we can find the zeros of functions

(25) we can replace by some approximate quadrature formulas, such as rectangles at *N* points on , and since is a circle, then to calculate quantities , 0,1,2, *<sup>k</sup> s k* , we obtain

> *<sup>f</sup> <sup>j</sup> s i N f N*

 

<sup>1</sup> <sup>2</sup> ( ) ( ) exp , ( )

*j j*

0 0 0 <sup>2</sup> exp , 1,2, , . *<sup>t</sup> j t <sup>j</sup> r ijs s* 

*k j*

 

1 ( ) 2 () *<sup>f</sup> m s <sup>d</sup> i f*

 

 

 *have in the region G m zeros* 1 2 , , , *<sup>m</sup>*

(23)

(24)

(25)

in the region *G* , such as a circle with center at

, you can find all the eigenvalues of problem (22),

. The integrals in (23) and

(27)

**/** , and applying the above statement to the

*t t c d* **[ ]** 

. The system itself (24) we solve using Newton's method, by

(26)

 

  *(with* 

0

*j k*

*k*

1

 

*k jt*

choosing the initial approximation on the border of the region *G* :

 

*N*

*<sup>f</sup> <sup>s</sup> d k i f*

1

*j*

*k*

*t t c d* **[ ]** 

belonging to the given region *G* , i.e. to the given interval ,

 **/** and radius ( )2 *t t t dc* 

**T**

2

 

(0)

*<sup>m</sup> <sup>k</sup>*

$$
\lambda\_{2l+1} = \lambda\_{2l} - \frac{f(\lambda\_{2l})f'(\lambda\_{2l})}{f'(\lambda\_{2l})^2 - f(\lambda\_{2l})f''(\lambda\_{2l})}
$$

$$
\lambda\_{2l+2} = \lambda\_{2l+1} - \frac{f(\lambda\_{2l+1})}{f'(\lambda\_{2l+1})}, \qquad l = 0, 1, 2, \dots \tag{29}
$$

The argument principle (23) and formula of the argument principle type (24), (25) were repeatedly applied in solving various spectral problems (see, for example, [1, 7, 9] and references therein), but the peculiarity of the proposed algorithm is to compute the values of function *f*( ) and its derivatives basing on *LU*-decomposition of the matrix ( ) *<sup>n</sup>* **T** .
