Non-Linear Simultaneous Equations Chapter 8 Solving Simultaneous Equations Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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: 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: 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 non-linear problems; multiple solutions are possible The solution found often depend on the initial guess entered before running Solver Engineering Computation: An Introduction Using MATLAB and Excel

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 Allowable x-values 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 Engineering Computation: An Introduction Using MATLAB and Excel

Simultaneous Equation Example Zooming in, we can see the solutions that we found with Solver 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 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 non-linear equations Important to recognize that multiple solutions may exist for non-linear problems Engineering Computation: An Introduction Using MATLAB and Excel