ChBE 354 Useful MATLAB Functions

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1 ChBE 354 Useful MATLAB Functions Assignment Statements a = 10 Stores 10 in a a = solve Hf HxL - f HzL, zl solves the equation for z and stores it in a a = single H143 ê 91L returns a single precision decimal approximation to the fraction. a : 10D will create the vector H1, 2, 3, 4, 5, 6, 7, 8, 9, 10L a : 2 : 10D will create the vector H0, 2, 4, 6, 8, 10L >> a : 2; 2 : 3D a = Creating Functions There are several ways to create functions. 1. m-file functions click on the blank sheet in the upper left hand corner. function fname = fx(x1) fname = x1^2; Calling this function a = fx(4) returns a value in a of Anonymous functions This is a new way of creating a function without using a m-file. On the command line type fname function fzr=@(r,r,omega) omega^2/(4*g)*(r^2-2*r^2); It is called as a = fzr(rr,rr,om) and returns the value of the function in a

2 2 MATLABNotes.nb Using solve solve can be used to solve equations symbolically or numerically. MATLAB provides faster ways to numerically solve equations. An example of a symbolic solution is shown below. The equation must be cast in the form of f(x) = 0, but it is not necessary to add the = 0 x1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 - x1 x2 - x1 = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ If x1 is known, solve for x2 1 - x2 >> syms x1 x2 >> solve H ' x1 ê H1 - x1l - Hx2 - x1l ê H1 - x2l', x2l ans = 2 * x1 - x1 ^2 or >> syms x >> solve H ' x ^ n + 2 * x - 4 ', xl ans = 5 ^H1 ê 2L ^ H1 ê 2L or >> f1 = ü HxL x ^2 + 2 * x - 4; >> solve Hf1 HxL, xl ans = 5 ^H1 ê 2L ^ H1 ê 2L

3 MATLABNotes.nb 3 TI-89 Calculator Solution for the first solve example F4 1 Define fc Hx1, x2, nl = Hx2 - x1l ê H1 - x2l ^ n F2 1 solve Hfc H0, 0.96, 1L = fc H0.96, x2l, x2l x2 =.9984 Using fzero fzero is a numeric routine that can be used to find the roots of an equation. It is normally fast and efficient. There is also a function called fsolve that uses the same syntax that can be used for solving nonlinear equations. fzero also assumes that the equation is equal to zero. It uses the function name and an initial guess as arguments. Solving the polynomial equation from a previous example f1 = ü HxL x ^ * x - 4; >> fzero Hf1, 0.2L ans = >> fzero Hf1, -2L ans = >> xr : 5D xr = >> f1 = ü HxL x.^ * x - 4; Note the. before the ^ this is necessary to insure that the function is evaluated at each value of xr. >> plot Hxr, f1 HxrLL >> grid on This example shows the importance of plotting functions before looking for the roots.

4 4 MATLABNotes.nb Plotting plot Hx, y, ' o 'L Will plot the values of y versus x using o for the points. Two sets of data can be plotted on the same plot using plot Hx, y, ' o ', x1, f1 Hx1LL Will plot (x,y) as points and (x1,f1(x1)) as a line. See the previous example for plotting a function. Curve Fitting The function polyfit can be used to fit a set of (x,y) data to a polynomial of degree n. The syntax is coef = polyfit Hx, y, nl polyfit returns the coefficients of the polynomial in coef (or any name that you want to assign). To evaluate the polynomial at any value of x use polyval. The syntax is ycal = polyval Hcoef, xl In this case, polyval returns a single value for the function. To generate a table of (x,ycal) replace the single value of x with a vector containing the values of x that you want ycal values calculated. The following was created using the f1 function from above and adding some random error. >> p = polyfit Hx, ynew, 2L p = >> plot Hx, ynew, ' o ', x, polyval Hp, xll

5 MATLABNotes.nb 5 Integration Matlab uses a version of Maple for some of the symbolic and numerical integration. There is also a polynomial integration function built into Matlab. Going back to the section on curve fitting, if a series of (x,y) data is available it can sometimes be fit to a polynomial using polyfit pfit=polyfit(x,y,3) Once the coefficients (pfit) are available, the integral of the polynomial can be found by using polyint pint=polyint(pfit) To evaluate the integral between x-values use polyval area=polyval(pint,x2)-polyval(pint,x1) A function can be integrated between limits using quad or quadl (quadl handles singularities better than quad) ans = quad(f(x),x1,x2) Differentiation A polynomial can be differentiated by using polyder pder=polyder(pfit) To evaluate the derivative at some value of x use polyval dx = polyval(pder,x) where x is a number The Matlab help files are useful, but you may have to try several search terms to find the information that you need.