Non-Linear Simultaneous Equations
Chapter 8
Solving Simultaneous
Equations
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Review: Linear Equations in Matrix Form
• The first step in using matrix methods to solve a
series of linear simultaneous equations is to write
them in matrix form
• For n simultaneous equations and n unknowns:
where A is the coefficient matrix (n × n); X is the matrix of
unknowns (n × 1), and C is the constant matrix (n × 1)
Engineering Computation: An Introduction Using MATLAB and Excel
Review: Linear Simultaneous Equations
• Recall that if there are more unknowns then
equations, then we cannot find a unique solution
• If there are more equations than unknowns, then
some equations must be redundant
• If there are exactly the same number of equations
and unknowns, then there may be a unique
solution. In this case the coefficient matrix will be
square
Engineering Computation: An Introduction Using MATLAB and Excel
Solution of System of Linear Equations
• We can find the unknown variables by multiplying
the inverse of the coefficient matrix by the
constant matrix
Engineering Computation: An Introduction Using MATLAB and Excel
Summary – Linear Equations
• If the inverse of the coefficient matrix exists, then
there is a solution, and that solution is unique
• If the inverse does not exist, then there are two
possibilities:
– The equations are incompatible, and so there are no
solutions, or
– At least two of the equations are redundant, and so
there are more unknowns than unique equations.
Therefore, there are an infinite number of solutions
Engineering Computation: An Introduction Using MATLAB and Excel
Non-Linear Equations
• If any of the equations are non-linear, then the
matrix method will not work
• Example: Consider these two equations:
• The x2 term in the first equation prevents the use
of a matrix solution
Engineering Computation: An Introduction Using MATLAB and Excel
Excel Solver
• Solver is an Excel Add-In that can find solutions to
many problems with multiple independent
variables
• The first step is to check that Solver is installed
• If so, it will appear in the Data group of the ribbon:
Engineering Computation: An Introduction Using MATLAB and Excel
Installing Solver
• If Solver is not present on
the ribbon, start by
clicking the Office
Button…
• And selecting Excel
Options…
Engineering Computation: An Introduction Using MATLAB and Excel
Installing Solver
• Choose Add-Ins…
• And click Go beside
Excel Add-Ins…
Engineering Computation: An Introduction Using MATLAB and Excel
Installing Solver
• Check Solver and click OK
• If Solver has not been installed, click Yes to install
it now
Engineering Computation: An Introduction Using MATLAB and Excel
Using Solver
• Solver attempts to set the value of a target cell to
its minimum, its maximum, or a specific value,
• By changing one or more input cells (independent
variables,
• While maintaining specified constraints (optional)
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• Consider this equation:
• Use Solver to find values of x for which
– y is minimized
– y is equal to exactly 50
– y is maximized, within limits of x from 0 to 10
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• To better understand this problem, consider this
plot of y vs. x:
200
180
y
160
140
120
100
80
60
40
20
0
-10
-5
0
5
10
15
x
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• Label a cell for the independent variable x and
enter the formula for y in another cell:
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• Start Solver. For the first part of the problem we
want to minimize y (Target Cell C4) by changing x
(cell C2):
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• The answer is that when x = 2, y = 16, the
minimum value of y that is possible:
200
180
y
160
140
120
100
80
60
40
20
0
-10
-5
0
5
10
15
x
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• Next, find the value of x for which y = 80:
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• But we can see from the
graph that there are two
solutions to the problem
• This is typical for nonlinear problems; multiple
solutions are possible
• The solution found often
depend on the initial
guess entered before
running Solver
200
180
y
160
140
120
100
80
60
40
20
0
-10
-5
0
5
10
x
Engineering Computation: An Introduction Using MATLAB and Excel
15
Solver Example
• Try x = -10 as the initial guess and run Solver
again:
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• Now try to maximize y:
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• Since the value of y increase to infinity for both
increasing and decreasing values of x, there are no
maxima values to be found
• However, we can constrain the problem by requiring
the input value of x to be between 0 and 10:
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• We find that when x = 10, y = 80. This is the
maximum value that y can have, subject to the
constraints on x.
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• However, even this
solution is affected by
the initial guess. For
example, if we try x =
-10 as the initial value:
• Then we get a different
solution, although the
value of y is clearly not
maximized:
Engineering Computation: An Introduction Using MATLAB and Excel
Solver Example
• It is important to realize that Solver finds local
maximum and minimum values
200
180
y
Allowable
x-values
160
140
120
100
80
60
40
20
0
-5
0
5
10
15
x
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• Back to our earlier example - consider these two
equations:
• Can we find values of x and y that satisfy both
equations?
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• With non-linear simultaneous equations, the first step
is to write the equations with all of the variables and
constants on the same side of the equal sign
• We give the resulting expressions variable names (f1
and f2 here):
• When both f1 and f2 are zero, then the equations are
satisfied
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• Start by labeling two cells for the input variables (x
and y)
• Label two cells for f1 and f2, and enter their
formulas:
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• Remember that Solver must have a single target
cell: we cannot specify that both f1 and f2 are to be
zero
• We could specify that the sum of f1 and f2 be zero;
however, this would not guarantee that both are
zero as one could be positive and the other
negative
• If we square both values, then both these values
must be greater than or equal to zero
• If the sum of the squares equals zero, then both f1
and f2 must be zero
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• Add cells for the squares and their sum. The sum
(cell C7) will be the Target Cell in the Solver setup:
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• Solver has found a
solution: x is about
-1 and y is about 0
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• These are the exact values for the solution:
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• Try different guess values of x and y:
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• A second solution if found: x = 1.5 and y = 12.5
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• A graphical solution is possible with two
equations. Notice that the curves intersect in two
places
80
60
Equation 1
y
Equation 2
40
20
0
-20
-40
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• Zooming in, we can see the solutions that we found
with Solver
20
15
Equation 1
y
Equation 2
10
5
0
-5
-10
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• Now try these equations (only the second is
changed)
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• In this case, no solution can be found
Engineering Computation: An Introduction Using MATLAB and Excel
Simultaneous Equation Example
• Graphing the equations shows that they do not
intersect
20
15
Equation 1
y
Equation 2
10
5
0
-5
-10
-15
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x
Engineering Computation: An Introduction Using MATLAB and Excel
Summary
• Excel Solver can be used to numerically find
maxima, minima, and specific values of a target
cell by varying one or more input cells, with or
without additional constraints
• Solver can also find solutions of simultaneous nonlinear equations
• Important to recognize that multiple solutions
may exist for non-linear problems
Engineering Computation: An Introduction Using MATLAB and Excel