Equations And Inequalities Algebraic Equations 11-17 “MATHEMATICS I” .
Systems of Nonlinear Algebraic Equations
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Transcript of Systems of Nonlinear Algebraic Equations
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Systems of Nonlinear Equations
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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
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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.
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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
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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
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Only one non-linear equation (eqn 4) must be solved. All others calculated directly
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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.
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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
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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.
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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
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Iterative Methods without Partial Derivatives
Gauss-Jacobi Similar to method of direct substitution
Example
Simultaneous Nonlinear Algebraic Equations
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481
33
21
33
121
23
22
21
322
1
1
xxx
xxxxxx
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Gauss-Jacobi
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*
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
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Spreadsheet for Gauss-Jacobi
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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
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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
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mixxxfx mii ,,2,1},,,,{ 21
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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.
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)1()(
_1()1(2
)1(1
)()(2
)(1
)()()(2
)(1
)1(
,,,,,,
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:,,2,1,1,,,
ki
ki
km
kki
km
kki
i
ii
kii
km
kkii
ki
xxxxxfxxxfs
st
wheremixtxxxftx
maxtti
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Wegstein example
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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
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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
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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
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0},,,{ 21 mi xxxf
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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)
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Newtons method
Equations can be written
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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
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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.
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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.
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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.
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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
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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
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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.
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Example of Excel Solver
Again, consider
Rearrange to
Simultaneous Nonlinear Algebraic Equations
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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
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Excel Solver
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Excel Solver
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Excel Solver
Solution obtained
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Matlab fsolve
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Matlab fsolve
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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.
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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.
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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.
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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
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End of Simultaneous non-linear algebraic equations