41 - 4/24/2000AME 150L1 Solving Engineering Problems, Integrating Equations

41 - 4/24/2000AME 150L2 HW 16 - “Sinc Function” The sinc function is common in signal processing (and the signal processing libraries on the student unix cluter have sinc defined)

41 - 4/24/2000AME 150L3 SINC Sin(pi*x)/(pi*x) function SINC(X) returns a matrix whose elements are the sinc of the elements of X, i.e. y = sin(pi*x)/(pi*x) if x ~= 0 = 1 if x == 0 where x is an element of the input matrix and y is the resultant output element.

41 - 4/24/2000AME 150L4 Fortran-ish sinc function y=sinc(x) for k=1:length(x); if x(k)==0 y(k)=1 else y(k)=sin(pi*x(k))/(pi*x(k)) end

41 - 4/24/2000AME 150L5 Dirty Matlab sinc function y=sinc(x) y=sin(pi*(x+2*eps))./(pi*(x+2*eps))    ||||| || ||||| ||||| note./ ||||| Small positive number

41 - 4/24/2000AME 150L6 Elegant Matlab sinc function y=sinc(x) y=ones(size(x)) index=find(x) y(index)=sin(pi*x(index))./... (pi*x(index)) This is used in the signal processing toolbox

41 - 4/24/2000AME 150L7 FIND Find indices of nonzero elements. I = FIND(X) returns the indices of the vector X that are non- zero. For example, I = FIND(A>100), returns the indices of A where A is greater than 100. See RELOP. [I,J] = FIND(X) returns the row and column indices of the nonzero entries in the matrix X. This is often used with sparse matrices. [I,J,V] = FIND(X) also returns a column vector of the nonzero entries in X. Note that find(X) and find(X~=0) will produce the same I and J, but the latter will produce a V with all 1's.

41 - 4/24/2000AME 150L8 Integrals of sin(x)/x quad('sync',0,2,eps)*pi quad8('sync',0,2,eps)*pi dx=x(2)-x(1);y=sync(x); trapz(y)*dx*pi dx= dx= dx= dx=1/8192= smallest possible size in student edition (16384 elements)

41 - 4/24/2000AME 150L9 Sounding Rocket A table of altitude vs. time for a sounding rocket is available Requirement: calculate the velocity and acceleration from the data Use the Matlab function diff

41 - 4/24/2000AME 150L10 DIFF Difference & approximate derivative DIFF(X), for a vector X, is [X(2)-X(1) X(3)-X(2)... X(n)- X(n-1)]. DIFF(X), for a matrix X, is the matrix of column differences, [X(2:n,:) - X(1:n-1,:)]. DIFF(X), for an N-D array X, is the difference along the first non-singleton dimension of X.

41 - 4/24/2000AME 150L11 diff for arrays DIFF(X,N) is the N-th order difference along the first non- singleton dimension (denote it by DIM). If N >= size(X,DIM), DIFF takes successive differences along dimension DIM until size(X,DIM) == 1. DIFF then takes any successive differences along the (M+1) dimension. DIFF(X,N,DIM) is the Nth difference function along dimension DIM. If N >= size(X,DIM), DIFF returns an empty array.

41 - 4/24/2000AME 150L12 Diff - examples h =.001; x = 0:h:pi; diff(sin(x.^2))/h is an approximation to 2*cos(x.^2).*x diff((1:10).^2) is 3:2:19 If X = [ ] then diff(X,1,1) is [ ], diff(X,1,2) is [ ], diff(X,2,2) is the 2nd order difference along the dimension 2, and diff(X,3,2) is the empty matrix.

41 - 4/24/2000AME 150L13 Interpolation Linear - table look-up Spline Interpolation –A Spline is an instrument draftsmen use Spline \Spline\, n. 1. … 2. A long, flexible piece of wood sometimes used as a ruler. Source: Webster's Revised Unabridged Dictionary, © 1996, 1998 MICRA, Inc.

41 - 4/24/2000AME 150L14 Cubic Spline Interpolation Fit a Cubic (4 adjustable constants) to three points, maintaining the slope as continuous as you pass to successive triads. –The curve passes through all of the data –The constants are fixed only over the triad of points –The slope (derivative) varies smoothly

41 - 4/24/2000AME 150L15 SPLINE Cubic spline interpolation YI = SPLINE(X,Y,XI) uses cubic spline interpolation to find a vector YI corresponding to XI. X and Y are the given data vectors and XI is the new abscissa vector XI. PP = SPLINE(X,Y) returns the pp-form of the cubic spline interpolant instead, for later use with ppval, etc.

41 - 4/24/2000AME 150L16 PPVAL Evaluate piecewise polynomial V = PPVAL(PP,XX) returns the value of the piecewise polynomial PP at the points XX. The piecewise polynomial form (pp-form) is returned by SPLINE.

41 - 4/24/2000AME 150L17 Example of a Spline Fit a cubic spline to a coarsely sampled sine » x = 0:10; y = sin(x); » xi = 0:.25:10; » yi = spline(x,y,xi); » plot(x,y,'o',xi,yi),grid,hold » dy=diff(yi)./diff(xi); » for k=1:length(xi)-1;xx(k)=(xi(k+1)+xi(k))/2;end; » plot(xx,dy,'r')