MATLAB Instructions

Matlab Instructions Matlab calculates with matrices and vectors and draws graphs in 2D and 3D. Skip the Intro­ duction and Help documents; as preliminary practice, just read and carry out the following. Entering matrices and vectors. In Matlab the variables represent matrices and vectors. The symbol = assigns the value on the right side of the equation to the symbol on the left. Type each of these lines in order, and see what you get. (Always hit [return] to end a line or command.) A = [1 2 3; 4 5 6; 7 8 9] (you can use commas instead of spaces: 1,2,3;) b = [5 2 1] b’ (transpose: gives the column vector which Matlab calls [5;2;1]) eye(3) (eye = I, the identity matrix) Try making a mistake: C = [1,2,3; 4,5]. To edit the mistake, press any of the four arrow keys to get the line back. (You can also prepare your commands in a text editor such as emacs and copy them with the mouse onto the Matlab command line.) Operations with matrices and vectors Sum, difference A+B, A-B (matrices must be same size) Product A*B (matrices must be compatibly sized) Powers A^n (A times itself n times; A must be square) Transpose A’ Inverse inv(A) (or A^-1) Try typing (use the values of A and b above): A+eye(3) 1 A*b A*(b’) A*b’ b*A Graphing with Matlab Array operations. Recall that * and ^ are product and power operations for matrices. Adding a dot before * or ^ makes these operations act component-wise. So, if x = [x 1 x2 . . . xn ], then exp(x) = [exp(x1 ) . . . exp(xn )] (similarly with sin, cos, log, etc.) x+y = [x1 + y1 . . . xn + yn ] (similarly with -) x.*y = [x1 y1 . . . xn yn ] m x.^m = [xm (m can be zero) 1 . . . xn ] Colon operator. This generates a vector with equally spaced entries; for example, [0 : 2 : 12] = [0 2 4 6 8 10 12]; [2 : −.1 : 1.6] = [2.0 1.9 1.8 1.7 1.6] 2D plot directions. Given x = [x1 x2 . . . xn ], y = [y1 . . . yn ] , plot(x, y) plots the n points (xi , yi ), joined by solid line segments. plot(x, y,� −−� ) plot(x, y,� �� ) plots the n points, joined by dashed line segments. plots the n points as individual stars (or dots or circles, etc). hold toggles between on and off (at the start it’s off ); when off, a new plot erases the previous one; when on, the new plot is superimposed on the old one. print gives a print-out of the current screen plot. Try in order (press [return] after each command): x=[0:.1:2] plot(x,sin(x)) plot(x,cos(x),’*’) hold plot(x,sin(x),’--’) hold plot(x,4*x.^3) (this plots y = 4x3 ; note the need for the array operator) You can also put graphs and scatter plots together without the hold command. The commands below graph the three functions 10x, 10x 1/2 , 2x5/3 . (With the semicolon at the end of each command Matlab won’t print out all the numbers. The semicolon also permits you to put several commands on one line.) x = [2:40:400]; w = [1:1:500]; plot(x,10*x, ’*’,w,b,w,c, ’--’); b = 10*(w.ˆ.5); 2 c = 2*(w.ˆ(5/3)); Graphing with Matlab (continued) 3D Plot directions. To plot z = f (x, y), you specify: the grid (xi , yj ) of lattice points: give the vectors x = [x 1 . . . xn ] and y = [y1 . . . yn ]. Example: To make a grid with spacing .1, over the interval [−2, 2] on both axes, type (in what follows, � is the matlab prompt; don’t type it — type the semicolon at the end so Matlab won’t print out all the numbers — remember [return] at the end) � x = [−2 : .1 : 2]; � y = [−2 : .1 : 2]; � [x, y] = meshgrid(x, y); the function z = f (x, y) For example, to graph the function f (x, y) = y 2 − x2 , type � z = y.ˆ 2 − x.ˆ 2; plot the graph either as a mesh of lines, or as a filled-in surface (the color indicates the value of z, i.e., the height of the graph above the xy-plane); type first � mesh(x, y, z) then � surf(x, y, z) change the viewpoint The default picture is shown at the right; to change the viewpoint (rotate left-right, or up-down), type � rotate3d then place the mouse cursor in the graph region, hold down left button, move mouse, release button. The two numbers on the screen are the azimuth: angle in degrees from the negative y-axis to the line of sight, and the elevation, the angle in degrees from the xy-plane to the line of sight. To turn off rotation, type again: � rotate3d hidden lines Try typing: � hidden changing scale [−20, 20], type (type it again to change back) To change the x-axis scale to [−4, 4], the y-axis to [−5, 5], and the z-axis to � axis([−4 4 − 5 5 − 20 20]) contour curves To get a 2D plot of level curves or a 3D plot with 20 contour curves, type � contour(x, y, z, 20) � contour3(x, y, z, 20) 3