powered by

SOLVING NONLINEAR SYSTEMS IN SCILAB
Everyday engineers encounter steady-state nonlinear problems in their real-caseapplications.
In this tutorial we show how nonlinear systems can be easily solved using Scilab.
Level

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

www.openeering.com

Step 1: Purpose of this tutorial
It is very common in the engineering area to solve steady state nonlinear
problems.
{Typically, two kinds of nonlinear systems arise:Systems withnonlinear equations inSystems withnonlinear equations in(Systems withunknowns.

(())

()
nonlinear equations inunknowns)

unknown

If the reader is interested in determining the zeros of polynomials, pleaserefer to the help of the main Scilab commands for managing polynomials(e.g. roots, poly, and horner).

{(Systems with

(())

()
nonlinear equations inunknowns)

Step 2: Roadmap
In the first part of this tutorial we show how to use the command fsolvefor equations and systems of equations. The command is used for solving
systems with exactly the same number of equations and unknowns.
The second part focuses on the use of the command lsqrsolve. In thislast part the reader can see how to solve systems with fewer unknowns
than equations.

Descriptions
fsolvelsqrsolve
Exercise
Final remarks and references

Steps
3-7
8-10
1112-13

Nonlinear systems in Scilab
www.openeering.com

page 2/12

Scilab syntax for fsolve:

Step 3: The fsolve function
The “fsolve” function solves systems ofunknowns.

[x [,v [,info]]]=fsolve(x0,fct [,fjac] [,tol])nonlinear equations andArguments:

The algorithm characteristics are:
The fsolve function is based on the idea of the Newton method;It is an iterative method, i.e. it starts from an initial approximation
value
and then it performs an iteration, obtaining , and so on;Only one solution is found by the command and this solution
depends on the initial approximation(basin of attraction).








x0: real vector (initial value of function arguments);
fct: external (i.e. function or list or string);
fjac: external (i.e. function or list or string);
tol: real scalar, precision tolerance: termination occurs when thealgorithm estimates that the relative error between x and the solution
is at most tol. (tol=1.d-10 is the default value);
x: real vector (final value of function argument, estimated zero);v: real vector (value of function at x);
info: termination indicator:
0: improper input parameters;
1: algorithm estimates that the relative error between x andthe solution is at most tol;
2: number of calls to fct reached;
3: tol is too small. No further improvement in theapproximate solution x is possible;
4: iteration is not making good progress.

For examples and more details see:
http://help.scilab.org/docs/5.3.0/en_US/fsolve.html

Nonlinear systems in Scilab
www.openeering.com

page 3/12

Step 4: fsolve example (scalar case)In this example we want to solve the following function;( )or, equivalently, to find the zero of( )( )The graphical representation is given in the following figure.

// Example 1deff('res=fct_1(x)','res=cos(x)-x')x0 = 0.
xsol =fsolve(x0,fct_1)
x = linspace(-2,2,51)
fcos = cos(x)fx = xscf(1)
clf(1)
plot(x,fcos,'r-');p = get("hdl"); p.children.thickness = 3;plot(x,fx,'b-');p = get("hdl"); p.children.thickness = 3;
(Coding of the example)
The obtained solution is: xsol = 0.7390851

Nonlinear systems in Scilab
www.openeering.com

page 4/12

-

Step 5: fsolve example (2-dimensional case)// Example 2deff('res=fct_2(x)',['res(1)=x(2)(x(1).^2+1)';'res(2)=x(1)-(2*x(2)-x(2).^2)/3'])
scf(2)
clf(2)
x1 = linspace(-3,3,101)
y1 = x1.^2+1

In this example we solve the following 2-dimensional problem{()

The graphical representation is given in the following figure. The figure
reveals the presence of two solutions. Depending on the initial point, thefunction fsolve can reach the first or the second solution as reported by
the example.

y2 = linspace(-3,5,51)x2=(2*y2-y2.^2)/3
plot(x1,y1,'r-');p = get("hdl"); p.children.thickness = 3;plot(x2,y2,'b-');p = get("hdl"); p.children.thickness = 3;x0 = [1;0]xsol1 =fsolve(x0,fct_2)
res1 = fct_2(xsol1)
x0 = [-3;8]xsol2 =fsolve(x0,fct_2)
res2 = fct_2(xsol2)(Coding of the example)
The obtained solutions are xsol1 = [0.3294085; 1.10851]
 xsol2 = [- 1.5396133; 3.3704093]
with



res1 = 10^(-14) * [ 0.1332268; 0.0610623]
res2 = 10^(-13) * [- 0.3375078; - 0.1310063]

Nonlinear systems in Scilab
www.openeering.com

page 5/12Step 6: fsolve example with embedded solver
In this example we combine the use of the fsolve function to solve aboundary value problem using the shooting method.
The idea is to embed the Ordinary Differential Equation (ODE) solver(shooting method) inside the fsolve function creating an appropriate
function to be solved. This approach is quite general since the ODE
solver can be seen as a black box function

The problem under consideration is the following:()( )( ) withand( ).

This boundary value problem can be reduced to the following initialdifferential equation problem{

̇ ( )

( )

̇ ( )( )where one of the initial condition in unknown, i.e.
{( )
( )

In order to use the fsolve function we need to introduce the function forwhich we want to find the zero. In this case, we define the function as( )((Example of black-box))( ) is the solution of the boundary value problem subject towhere
the initial condition that depends on .

Nonlinear systems in Scilab
www.openeering.com

page 6/12

Step 7: Coding and solving
The Scilab code is reported on the right. In the following figures we report
the initial solution and the optimal solution.

(Initial solution with

Changing the initial point
solution.

and optimal solution withto)// Example 3function wdot=wsystem(t, w)
wdot(1) = w(2)wdot(2) = 3/2*w(1).^2
endfunctions = -1
w0 = [4;s]t0 = 0t = linspace(0,1)// winit = ode(w0,t0,t,wsystem);// plot(t,winit)
deff('res=fct_3(s)',['w0 = [4;s];','w =ode(w0,t0,t,wsystem);','res=w(1,$)-1'])
s = -1ssol =fsolve(s,fct_3)

it is possible to find a different// compute solution
w0 = [4;ssol]t0 = 0
t = linspace(0,1)
wsol = ode(w0,t0,t,wsystem);plot(t,wsol)p = get("hdl"); p.children.thickness = 3(Example’s code)(Initial solution withand optimal solution with)

Nonlinear systems in Scilab
www.openeering.com

page 7/12

Step 8: Nonlinear least square fitting
Nonlinear least square is a numerical technique used when we havenonlinear equations in
unknowns. This means that, in these cases,
we have more equations than unknowns. This case is very common andinteresting and it arises, for example, when we want to fit data withnonlinear (and non polynomial) equations.

The mathematical problem corresponds to find a local minimizer
following equation( )of the∑( ( ))where n is the number of points, and f i represents the residual betweenth
the i given measure data point and the interpolation model (estimateddata). Reducing the total sum of residuals corresponds to find the optimal
parameters for our fitting model with our given data.
For example,interpolations.

thisisparticularly

usefulforexperimental

data(Example of nonlinear fitting)Nonlinear systems in Scilab
www.openeering.com

page 8/12

Scilab syntax for lsqrsolve:

Step 9: lsqrsolve
In Scilab the solution of problems in which we have more equations thanunknowns is obtained using the lsqrsolve function.

[x [,v [,info]]]=lsqrsolve(x0,fct,m [,stop [,diag]])[x [,v [,info]]]=lsqrsolve(x0,fct,m ,fjac [,stop [,diag]])

Arguments:

The algorithm features are:

The lsqrsolve function is based on the Levenberg-Marquardt
algorithm;

If the Jacobian is not provided it is calculated by a forwarddifference approximation;

It is an iterative method, i.e. it starts from an initial approximation
value
and then it performs an iteration obtaining , and so on;Only one solution is found and this solution depends on the initialapproximation(basin of attraction).











x0: real vector of length n (initial value of function argument);
fct: external (i.e. function or list or string);
m: integer, the number of functions. m must be greater than or equal
to n;fjac: external (i.e. function or list or string);
stop: optional vector[ftol,xtol,gtol,maxfev,epsfcn,factor] the default valueis [1.d-8,1.d-8,1.d-5,1000,0,100]
diag: is an array of length n. diag must contain positive entries thatserve as multiplicative scale factors for the variables;
x: real vector (final estimate of the solution vector);v: real vector (value of fct(x));
info: termination indicator

For examples and more details see:
http://help.scilab.org/docs/5.3.3/en_US/lsqrsolve.html

Nonlinear systems in Scilab
www.openeering.com

page 9/12

Step 10: lsqrsolve example
In this example we want to estimate the parameter of the following
Gaussian function (fgauss)()over a set of 100 data points. The data points are generated starting froma Gaussian distribution, adding a uniform noise. The obtained result isreported in the following figure.
We will have one equation for any point, so the total number of equations
is equal to 100. In our code the error function is named fgausseq.

// Example 4function y=fgauss(x, A, x0, sigma)
y = A*exp(-(x-x0).^2/(2*sigma^2))
endfunction
xdata = linspace(0,1)';
A = 1.0;x0 = 0.5;
sigma = 0.1;
ydata = fgauss(xdata, A, x0, sigma) +(rand(xdata)-0.5)*0.1;// plot(xdata,ydata)
function err=fgausseq(param, m)
A = param(1);x0 = param(2);sigma = param(3);err = ydata -fgauss(xdata, A, x0, sigma);
endfunction
pinit = [0.25; 0.25; 0.25][psol,v, info] =lsqrsolve(pinit,fgausseq,size(xdata,1))
disp(psol)plot(xdata,ydata,'ko')p = get("hdl"); p.children.thickness = 3
plot(xdata,fgauss(xdata, psol(1), psol(2),psol(3)),'b-')
p = get("hdl"); p.children.thickness = 3(Example’s code)(Optimal solution and given data)

Nonlinear systems in Scilab
www.openeering.com

page 10/12

Step 11: Exercise #1
Modify the previous example considering the following distribution(Student's t-distribution)
( )

(√)
( )

Here, we are interested in estimating()from measured data.

Compare t-student and Gaussian distributions.

(Initial distribution)

Hints: The gamma function in Scilab is gamma.

(Comparisons between two different distributions)Nonlinear systems in Scilab
www.openeering.com

page 11/12

Step 12: Concluding remarks and References
In this tutorial we have shown how to solve nonlinear problems in Scilab.
We described the use of both the fsolve and lsqrsolve functions.
On the right-hand column you may find a list of interesting references forfurther studies.

1. Scilab Web Page: www.scilab.org
2. Openeering: www.openeering.com
3. http://help.scilab.org/docs/5.3.0/en_US/fsolve.html
4. http://en.wikipedia.org/wiki/Shooting_method
5. http://help.scilab.org/docs/5.3.3/en_US/lsqrsolve.html

Step 13: Software content
To report bugs or suggest improvements please contact the Openeering
team at www.openeering.com.
--------------------------Nonlinear systems in Scilab---------------------------------------Main directory-------------ex_nonlinear.sce
ex1.scelicense.txt: All the examples: Solution of exercise 1: The license fileThank you for reading,
Manolo Venturin and Silvia Poles

Nonlinear systems in Scilab
www.openeering.com

page 12/12