Systems of Nonlinear Algebraic Equations

Systems of Nonlinear Equations

Systems of Nonlinear A general system of n nonlinear equations:

In general, it is difficult to solve such sets of equations Bounding techniques cannot be used. In addition, plotting a function to locate the solutions,

which will work for a single nonlinear equation, will not generally work for a system of nonlinear equations.

Use algorithms to minimize the number of equations that must be solved.

1 2( , , ..., ) ( ) 0 ( 1, 2,..., )n iif x x x f i n x

Simultaneous Nonlinear Algebraic Equations

2

Precedence Ordering Reduce the number of equations that must

be solved simultaneously Identify two classes of variables and

appropriately order equations: Equations that can be solved for an unknown

variable irrespective of the other equations solve these first

Variables that appear in only one equation. Solve these equations after all others have been solved.


3

ExampleEquation Nu

(Number of unknowns)

x1x2 + x6x4 = 18 4 (1)x2 + x5 + x6 = 12 3 (2)

x xx1

2

4

3 ln

3 (3)

x x32

3 2 1 (4)x2 + x4 = 4 2 (5)x3 (x3 + x6) = 7 2 (6)

1. Solve equation 4 for x3 and remove variable from set.Simultaneous Nonlinear Algebraic

Equations5

Apply algorithmSolution Order Output variable Equation

1 x3 x x32

3 2 2 x6 x

xx6

33

7

3 x4 184

4 36 44

4

4

x xx

xx

ln

4 x1 x x xx16 4

4

184

5 x2 x2 = 4 - x4

6 x5 x5 = 12 - x2 - x6


8

Only one non-linear equation (eqn 4) must be solved. All others calculated directly

The Tearing Method Is useful when most of the nonlinearity of a problem is

associated with a single unknown. The nonlinear variable and one nonlinear equation are

torn from the problem and a value for the nonlinear unknown is assumed.

The remaining equations are solved for the remaining unknowns using any desired method.

The values of the calculated unknowns are combined with the assumed value of the nonlinear unknown to evaluate the torn equation.

Adjust the nonlinear unknown until the torn equation is satisfied thus solving the original problem.


9

Conclusions Tearing and Precedence ordering algorithms

can reduce the number of equations that must be solved simultaneously

These algorithms can increase the potential for finding a solution

We will focus on techniques for solving the equations, but recall that these algorithms will increase the potential for solving a problem


12

Numerical Methods All methods implement iterative solution

techniques. Each method has another way to update the

solution. All methods may sometimes converge

toward a solution and may also diverge, depending on Specifics of the problem The initial guess used.


13

Three broad classes of methods Methods that dont use partial derivatives

Gauss-Jacobi like direct substitution Wegstein like false position

Methods that use partial derivatives Analytical partial derivatives Partial derivatives based on finite difference

estimates Optimization-based methods

Excels solver Matlab fsolve function


14

Iterative Methods without Partial Derivatives

Gauss-Jacobi Similar to method of direct substitution

Example


15

481

33

21

33

121

23

22

21

322

1

1

xxx

xxxxxx

Gauss-Jacobi


16

*

2

32

1*3

*3

*2

*131

212*

32*

1*3

*2

*122

*2

21*

1*3

*2

*113

4,,

81,,

33,,

x

xxxxfx

xxxxxfx

xxxxxfx

Rearrange to obtain an equation for each variable

Iterate using these equations, check for convergence by comparing change in each variable from one step to the next, e.g. for the kth iteration,

11

111

1 k

kk

xxxerror

Spreadsheet for Gauss-Jacobi


17

k x1 x2 x3 error1 error2 error31.0000 2.0000 10.0000 5.00002.0000 0.5488 7.2111 3.1586 -0.726 -0.279 -0.3683.0000 1.5229 8.4096 4.4735 1.775 0.166 0.4164.0000 0.7964 7.6595 3.7773 -0.477 -0.089 -0.1565.0000 1.1354 8.1300 4.1919 0.426 0.061 0.1106.0000 0.9157 7.8828 3.9280 -0.194 -0.030 -0.0637.0000 1.0427 8.0457 4.0649 0.139 0.021 0.0358.0000 0.9704 7.9617 3.9747 -0.069 -0.010 -0.0229.0000 1.0144 8.0163 4.0211 0.045 0.007 0.012

10.0000 0.9901 7.9876 3.9910 -0.024 -0.004 -0.00711.0000 1.0049 8.0057 4.0068 0.015 0.002 0.00412.0000 0.9967 7.9960 3.9968 -0.008 -0.001 -0.00213.0000 1.0017 8.0020 4.0022 0.005 0.001 0.00114.0000 0.9989 7.9987 3.9989 -0.003 0.000 -0.00115.0000 1.0006 8.0007 4.0007 0.002 0.000 0.00016.0000 0.9996 7.9996 3.9996 -0.001 0.000 0.00017.0000 1.0002 8.0002 4.0002 0.001 0.000 0.00018.0000 0.9999 7.9999 3.9999 0.000 0.000 0.000

Methods without partial derivatives

Wegsteins Method Similar to a multi-dimensional false position

method Generally requires fewer iterations that Gauss-

Jacobi and less likely to diverge Again write equations as Use direct substitution to obtain 2nd guess


18

mixxxfx mii ,,2,1},,,,{ 21

Wegstein method

Use iterative equations for subsequent guesses

We usually limit the distance you can extrapolate by setting A typical value for tmax is 10.


19

)1()(

_1()1(2

)1(1

)()(2

)(1

)()()(2

)(1

)1(

,,,,,,

11

:,,2,1,1,,,

ki

ki

km

kki

km

kki

i

ii

kii

km

kkii

ki

xxxxxfxxxfs

st

wheremixtxxxftx

maxtti

Wegstein example


20

k x1 x2 x3 f1 s1 t1 f2 s2 t2 f3 s3 t3 error1 error2 error3

1.0000 2.0000 10.0000 5.0000 0.5488 7.2111 3.1586

2.0000 0.5488 7.2111 3.1586 1.5229 -0.6712 0.5984 8.4096 -0.4298 0.6994 4.4735 -0.7141 0.5834 -0.726 -0.279 -0.368

3.0000 1.1317 8.0494 3.9257 1.0219 -0.8595 0.5378 8.0192 -0.4657 0.6823 3.9675 -0.6596 0.6026 1.062 0.116 0.243

4.0000 1.0727 8.0288 3.9509 1.0149 0.1189 1.1349 8.0150 0.2071 1.2611 3.9812 0.5427 2.1865 -0.052 -0.003 0.006

5.0000 1.0071 8.0113 4.0171 0.9922 0.3469 1.5313 7.9905 1.4004 -2.4974 3.9939 0.1916 1.2370 -0.061 -0.002 0.017

6.0000 0.9842 8.0634 3.9884 0.9965 -0.1872 0.8423 8.0078 0.3313 1.4954 3.9695 0.8468 6.5275 -0.023 0.006 -0.007

7.0000 0.9945 7.9802 3.8654 1.0544 5.6242 -0.2163 8.0666 -0.7071 0.5858 4.0103 -0.3312 0.7512 0.010 -0.010 -0.031

8.0000 0.9816 8.0308 3.9742 1.0058 3.7514 -0.3635 8.0151 -1.0170 0.4958 3.9858 -0.2247 0.8165 -0.013 0.006 0.028

9.0000 0.9728 8.0230 3.9837 1.0032 0.2941 1.4166 8.0115 0.4660 1.8725 3.9902 0.4670 1.8763 -0.009 -0.001 0.002

10.0000 1.0159 8.0014 3.9960 1.0013 -0.0444 0.9575 8.0000 0.5316 2.1349 3.9983 0.6567 2.9131 0.044 -0.003 0.003

11.0000 1.0020 7.9984 4.0028 0.9992 0.1550 1.1834 7.9984 0.5383 2.1661 4.0007 0.3512 1.5412 -0.014 0.000 0.002

12.0000 0.9987 7.9984 3.9996 1.0004 -0.3649 10.0000 8.0004 -45.5222 0.0215 4.0009 -0.0714 0.9333 -0.003 0.000 -0.001

13.0000 1.0158 7.9984 4.0008 0.9999 -0.0279 0.9729 7.9976 -63.7828 0.0154 3.9998 -0.8648 0.5363 0.017 0.000 0.000

14.0000 1.0003 7.9984 4.0003 1.0001 -0.0131 0.9871 7.9998 -180.0035 0.0055 4.0008 -1.8089 0.3560 -0.015 0.000 0.000

15.0000 1.0001 7.9984 4.0005 1.0000 0.3033 1.4354 7.9998 -7.8148 0.1134 4.0008 0.0564 1.0598 0.000 0.000 0.000







21

Newtons Method Also called Newton-Raphson method Write equations as Write a Taylor series expansion of each equation

about the guess value Rearrange the equation to solve for a new guess

value Newtons method applied to a general set of n

nonlinear equations is given by

1

1, 2, ,j

nj jk

i ki i x

fd f k n

x

x

x


22

0},,,{ 21 mi xxxf

Newtons method This equation generates a system of n linear

equations for which the coefficient matrix, called the Jacobian, is the partial derivatives of the nonlinear equations evaluated at the current values of the unknowns

The right hand side vector is a vector of the negative values of the function values evaluated at the current values of the unknowns and di=xi(j+1) -xi(j)


23

Newtons method

Equations can be written


24

mm

m

mmm

m

m

x

xxx

xf

xf

xf

xf

xf

xf

xf

xf

xf

2

1

3

2

1

**2*1

*

2

*2

2

*1

2

*

1

*2

1

*1

1

Newtons Method

The Jacobian used by Newtons method can be calculated using analytical expressions for the elements of the Jacobian or it can be calculated numerically using finite difference formulas.

The finite difference approach is easier to apply, but is susceptible to reliability issues for highly nonlinear systems of equations.


25

Newtons Method Newtons method will solve over 95% of

the nonlinear equation problems encountered in engineering.

Similar to the results for a single nonlinear equation, A good initial guess for the unknowns is many

times required for reliable convergence. The more linear that equations can be

formulated, the greater the reliability for convergence and the better the computational efficiency.


26

Newtons Method Davis provides a VBA macro QUASINEWTON

to solve a set of equations using Jacobian calculated by finite difference (see section 6.3.4, page 209)

Lecture notes show VBA and Matlab code to solve an example with analytically determined Jacobian

Lecture notes show Matlab code to solve example with Jacobian determined with finite difference approximation.


27







28

Optimization-based Techniques

Write problem as f(x) = 0. For any value of x other than the solution,

f(x) will not be zero. Could be positive or negative.

Write a new function Use a multidimensional optimization code

to minimize this function. If the minimum is equal to zero, then it solves the problem.

2)]([)( xfxF


29

Excel and Matlab tools

Excel solver implements optimization Matlab fsolve function performs

optimization of the same function. Lecture notes show implementation of these

tools. See section 2.8 on page 62 of Davis for how to turn on the Solver in Excel.

Excel solver and Matlab fsolve optimize the same function.


30

Example of Excel Solver

Again, consider

Rearrange to


31

481

33

21

33

121

23

22

21

322

1

1

xxx

xxxxxx

04),,(081),,(

033),,(

21

33

1213213

23

22

213212

322

1

13211

xxxxxxf

xxxxxxfxxxxxxf

Excel Solver


32

Excel Solver


33

Excel Solver

Solution obtained


34

Matlab fsolve


35

Matlab fsolve


36

Convergence and Accuracy

Iterative solutions are required; therefore, convergence is an issue. Similar to a single nonlinear equation, a relative convergence criterion is recommended. For convergence all the unknowns should satisfy the relative convergence criteria.

Accuracy is met by reducing the convergence criteria until the desired accuracy is attained.


37

Stability and Computational Efficiency

Divergence during the iterative solution process can be referred to as unstable convergence.

The computational efficiency is primarily related to the computational efficiency of the linear equation solver because Newtons method is based on solving a series of linearized problems.


38

Important issues

Ensure equations formulated properly. Make the equations as linear as possible Make sure you have reasonable starting values for the

unknowns. Physical considerations may provide a good starting value, as can simplifications of the nonlinear equations.

Ensure that the answers are accurate. Choose the proper convergence criterion and check your answers in the original functions.

Always ensure the solution makes sense from an engineering standpoint.


39

Conclusions

Solution of Simultaneous non-linear algebraic equations is difficult

Several tools at our disposal Always reduce the set of equations as much

as possible using precedence ordering or tearing methods.

Many can be solved with Newtons method


40

End of Simultaneous non-linear algebraic equations

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Systems of Nonlinear Algebraic Equations

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