Solving General Linear Equations w/ Excel

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Everett Hood
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1 Solving General Linear Equations w/ Ecel

2 Matri Operations in Ecel Ecel has commands for: Multiplication (mmult) matri multiplication Transpose (transpose) transpose a matri Determinant (mdeterm) calc the determinate Inverse (minverse) generate the matri inverse Important to remember that these commands apply to an array of cells instead of to a single cell When entering the command, you must identify the entire array where the answer will be displayed!

3 Practice Problem Find AB = C (2 X 4) (4 X 3) =?

4 Ecel Matri Multiplication First enter the two matrices (for A and B):

5 Ecel Matri Multiplication Good practice to label the matrices:

6 Ecel Matri Multiplication Shading and borders help the matrices stand out: A B = C.what size matri must C be?

7 Ecel Matri Multiplication An array of cells for the product must be selected in this case, a 2 3 array: Select the cells directly type mmult(array1,array2). You couldn t paste the equation into the selected cells, they are in different dimensions.

8 Ecel Matri Multiplication The MMULT function has two arguments: the ranges of cells to be multiplied. Remember that the order of multiplication is important.

9 Try this out! Ecel Matri Multiplication Using the Enter key with an array command only returns an answer in a single cell. Instead, use Ctrl + Shift + Enter keys with array functions (This only works when you highlight the solution area, then put the cursor in the top function typing area )

10 Ecel Matri Multiplication Answer cells formatted:

11 Try this out! Ecel Transpose (This only works when you highlight the solution area, then put the cursor in the top function typing area ) Use Ctrl + Shift + Enter to input command

12 Ecel Determinant Since determinant is a scalar, select a single cell and use Enter to input command

13 Ecel Matri Inversion Remember that only square matrices can have inverses

14 Ecel Matri Inversion Ctrl + Shift + Enter to input command

15 Try this out: make a matri get its inverse matri multiply these two matri to get the identity matri A X A -1 = I, the identity matri:

16 Solving Linear Equations - Ecel Consider these 3 linear equations with unknowns, y and z: Rewrite in matri form.

17 MATLAB solution: >> A=[ ; 0 2 3; ]; >> b=[-30; 11; 42]; >> =A\b =

18 Ecel Solution Enter coefficient and constant matrices: b =

19 Ecel Solution Label and highlight cells for matri of unknown variables: b =

20 Ecel Solution Ecel does not have the convenient left division operator, so we must enter = A -1 b. Enter formula to invert A matri and multiply the result by the b matri. This can be done in two steps or with nested commands: b =

21 Ecel Solution Apply formula to the selected array of cells by pressing Ctrl + Shift + Enter: b =

22 Eercises Solve the following sets of equations using MATLAB and Ecel and check the answer z y z z y

23 Solution 1 3 2y 4z z 2 5y 4z A = B = y z -4 =

24 Solution A = B = =

25 Other uses of A=b and left-division So far we have been given different equations for a set of variables and solved for those variables: Coefficients are known = = , 2 are unknown 3-3 It is common to have eperimental measurements where the data are known but the equation or the equation coefficients are unknown

26 Fitting eperimental data Suppose someone is eperimenting with a temperature controller. They apply a series of voltages and measure the resulting temperature. Voltage Temperature

27 Fitting eperimental data You have reason to believe the relationship should be linear, i.e. T = c 1 V + c 2 You know T, V You want to find c 1, c 2 Notice this is the same equation as y = m + b, so we re solving for slope and intercept.

Fitting eperimental data First, let s have a look at the data: >> V = [2:2:10]; >> T = [88.78, 95.79, 97.45, 107.43, 115.

28 Fitting eperimental data First, let s have a look at the data: >> V = [2:2:10]; >> T = [88.78, 95.79, 97.45, , ]; >> plot (V, T, * ); Not eactly a pretty straight line. Could be - The system really is not linear - The voltage wasn t eactly what you thought - The temperature measurement is not accurate enough - Etc.

29 Least Squares Fitting y What line best represents the data? Forcing the line through a data point is risky The best fit might not actually go through any points

30 Least Squares Algorithm y Least Squares minimizes the error between the line and the data points yi m i Find the line that minimizes the sum: i ( y i m i 2 ) Data point Line

31 Put our data into matri form Voltage Temperature We have one equation, c 1 V + c 2 = T, and 5 measurements: c 1 * 2 + c 2 = c 1 * 4 + c 2 = c 1 * 6 + c 2 = c 1 * 8 + c 2 = c 1 * 10 + c 2 = (m * + b = y) (m * V + b = T)

32 Put our data into matri form c 1 * 2 + c 2 = c 1 * 4 + c 2 = c 1 * 6 + c 2 = c 1 * 8 + c 2 = c 1 * 10 + c 2 = c1 c2 = Now it is in a familiar form, A=b

33 This is over-determined c1 c2 = Before we said we wanted equal number of equations and unknowns for a unique solution. Here we have 5 equations and 2 unknowns. This would be redundant if the points were all eactly on a line. Since they are not, this finds the best (least-squares) compromise.

34 This is over-determined c1 c2 = A b We can use the same left-division operator to solve for c1, c2. MATLAB recognizes the system is over-determined and switches to a least-squares fit algorithm. = A\b

35 Solve % least square fit clear all close all V=[2:2:10]; T=[88.78,95.79,97.45,107.43,115.85]; plot(v,t,'* ); A=[2 1;4 1;6 1;8 1;10 1]; TT=transpose(T); =A\TT; hold on line_vs=linspace(0,10,10); line_ts=line_vs*(1)+(2); plot(line_vs,line_ts); = Our vector components are c1, c2

36 Least squares fit to V, T data

37 Other function types Other types of functions can be fit using the same technique. Consider this data t y

38 Fitting a non-linear function A linear fit looks like this: You decide a decaying eponential function might be better

39 Fitting a non-linear function The equation we want to try is y(t) = c 1 + c 2 e -t Fitting a non-linear function does not mean the system of equations is non-linear. We will solve for c1, c2. This equation is still linear with respect to c1, c2.

40 Put into matri form c 1 + c 2 e -0 = 0.82 c 1 + c 2 e -0.3 = 0.72 c 1 + c 2 e -0.8 = e 0 1 e e e e e 2.3 c1 c2 =

41 Solve clear all close all A=[1 ep(0); 1 ep(-0.3); 1 ep(-0.8); 1 ep(-1.1);1 ep(-1.6); 1 ep(-2.3)]; b=[0.82;0.72;0.63;0.60;0.55;0.50]; 1=[0;0.3;0.8;1.1; 1.6; 2.3]; plot(1,b,'*');% plot the scattering dots hold on =A\b; line_=linspace(0,2.5,20); line_y=(1)+(2)*ep(-line_); plot(line_,line_y); = e 0 1 e e e e e 2.3 c1 c2 = y(t) = e -t A b

42 New fit

43 Summary Curve fitting is a commonly-needed operation and can be accomplished in Matlab as an A=b problem. MATLAB s left-division operator can be used to find an eact solution in the case of equal number of equations and unknowns. It can also be used to perform a least-squares fit in the case of more equations than unknowns. Be aware though that the underlying algorithm is not the same.

Name Class Date. subtract 3 from each side. w 5z z 5 2 w p - 9 = = 15 + k = 10m. 10. n =

Name Class Date. subtract 3 from each side. w 5z z 5 2 w p - 9 = = 15 + k = 10m. 10. n = Reteaching Solving Equations To solve an equation that contains a variable, find all of the values of the variable that make the equation true. Use the equalit properties of real numbers and inverse operations