The Chebyshev–Shamanskii Method for Solving Systems of Nonlinear Equations

Abstract

We present a method, based on the Chebyshev third-order algorithm and accelerated by a Shamanskii-like process, for solving nonlinear systems of equations. We show that this new method has a quintic convergence order. We will also focus on efficiency of high-order methods and more precisely on our new Chebyshev–Shamanskii method. We also identify the optimal use of the same Jacobian in the Shamanskii process applied to the Chebyshev method. Some numerical illustrations will confirm our theoretical analysis.

Appendices

Appendix A: Test Functions

We tested our algorithms on the following functions, from the Moré, Garbow and Hillstrom collection [1]:

$$ f(x)=\sum_{i=1}^{m} f_i^{2}(x), $$

(A.1)

where f ₁,…,f _m are defined by the following problems. We adopt the following general format described in [1] and add the expression of $F_{i}= \frac{\partial f(x)}{\partial x_{i}}$.

Name of function

• (a) Dimensions

• (b) Function definition

• (c) Standard starting point

• (d) Minima

• (e) Expression of the Gradient $F_{i}= \frac{\partial f(x)}{\partial x_{i}}$.

Thus, our four test functions are:

Problem 1

Broyden tridiagonal function

• (a) n variable, m=n

• (b) f _i (x)=(3−2x _i )x _i −x _i−1−2x _i+1+1 where x ₀=x _n+1=0

• (c) x ₀=(−1,…,−1)

• (d) f=0

• (e) F _i =−4f _i−1(x)+2(3−12x _i )f _i (x)−2f _i+1(x).

Problem 2

Broyden banded function

• (a) n variable, m=n

• (b) $f_{i}(x)=x_{i}(2+5x_{i}^{2})+1-\sum_{j \in J_{i}} x_{j}(1+x_{i})$ where J _i ={j:j≠i,max(1,i−m _l )≤j≤min(n,i+m _u )} and m _l =5, m _u =1

• (c) x ₀=(−1,…,−1)

• (d) f=0

• (e) $F_{i}=2(1+15x_{i}^{2}-2x_{i})f_{i}(x)$.

Problem 3

Discrete boundary value function

• (a) n variable, m=n

• (b) f _i (x)=2x _i −x _i−1−x _i+1+h ²(x _i +t _i +1)³/2 where h=1/n+1=0, t _i =ih and x ₀=x _n+1=0

• (c) x ₀=(ζ _j ) where ζ _j =t _j (t _j −1)

• (d) f=0

• (e) $F_{i}=-2f_{i-1}(x)+2[2+\frac {3}{2}h^{2}(x_{i}+t_{i}+1)^{2}]f_{i}(x)-2f_{i+1}(x)$.

Problem 4

Discrete integral equation function

• (a) n variable, m=n

• (b) $f_{i}(x)=x_{i}+h[(1-t_{i})\sum_{j=1}^{i} t_{j}(x_{j}+t_{j}+1)^{3}+t_{i}\sum_{j=i+1}^{n} (1-t_{j})(x_{j}+ t_{j}+1)^{3}]/2$ where h=1/(n+1), t _i =ih and x ₀=x _n+1=0

• (c) x ₀=(ζ _j ) where ζ _j =t _j (t _j −1)

• (d) f=0

• (e) $F_{i}=2[1+\frac {3}{2}ht_{i}(x_{i}+t_{i}+1)^{2}]f_{i}(x)$.

Table 4 gives details about the costs of the tested problems.

Table 4 Detailed intrinsic cost for the test functions, considering Eq. (A.1)

Appendix B: Unsymmetric Systems

If one considers unsymmetric systems, the detailed analysis for the costs involved in high-order methods is given in Table 5.

Table 5 Costs of operations involved in high-order methods for unsymmetric systems

Thus, the costs of the Newton and Chebyshev methods are:

Consequently, the efficiency of the Newton method is

and the efficiency of the Chebyshev method is

As in Sect. 5, we obtain the same ratio between efficiencies of the Chebyshev–Shamanskii method and the Chebyshev method, as the additional cost of the Chebyshev–Shamanskii method given in Corollary 3.1 is still valid, i.e.

Thus, if c is approximated by n ², E(CS)/E(d _C )>1 for n>22; in other words, the Chebyshev–Shamanskii method is more efficient than the pure Chebyshev’s for n>22. If c is approximated by n ³, then the same result is valid for n>29.