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newton_GGN11_sp

find the 0 of a function by Grau-Sanchez, Grau and Noguera (2011) method

CALLING SEQUENCE

[xs,fs,maxfs]=newton_GGN11_sp(x0,func,Jac,indJac,ftol,itermax,alphamin)

PARAMETERS

Input

* x0 = a (n x 1) vector of starting values

* func = a function, whose 0 is searched

* Jac = a function, performing the non zero values of the Jacobian of this function (with the rhs as an output)

* indJac = a (K x 2) matrix, collecting the non zero indexes of the Jacobian

* ftol = the convergence criterion for the function value

* itermax = the maxmimum number for the function calls

* alphamin = the minimum value used to dampen the step when convergence is not achieved

Output

* xs = the value of the parameter at solution

* fs = the value of the function at solution

* maxfs = the maximum value of the absolute function at solution

DESCRIPTION

Finds the 0 of an equation using the variant of the Newton method provided by Grau-Sanchez, Grau and Noguera "On the computational efficiency index and some iterative methods for solving systems of nonlinear equations", Journal of Computational and Applied Mathematics 236 (6), 1259-1266. Adpated to a problem when the Jacobian is sparse.

EXAMPLE

// Search the zero of function test_func in the optsci folder. The function codes the problem 7 provided by
// Sharma, Sharma and Bahl (2016), An improved Newton–Traub composition for solving systems of nonlinear equations,
// Applied Mathematics and Computation, N. 290, p. 98-110.
// The function to minimize is:
// x(i)x(i+1)=1 for i=1:98
// x(99)x(1)=1
// The corresponding Jacobian function is coded in function test_Jac_sp
// Starting values are the ones provided in the paper: -4*ones(99,1)
// Convergence criterion is set to 1e-10, maximum # of iterations is set to 1000, 
// minimum value used to dampen, if needed, the step is set to 0.05
 
ind_sp=[[1:99 1:99]' , [1:99 2:99 1]'];
[x0,f0,maxf0]=newton_GGN11_sp(-4*ones(99,1),newton_testfunc_sp,newton_testJac_sp,ind_sp,1e-8,100,0.05)
 
// Provides the expected result x0 =-1*ones(99,1) and f0=0

AUTHOR

Éric Dubois 2019

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