1. Mathematical Modeling and Simulation Interpolation and Curve Fitting Using MATLAB Prof. Muhammad Saeed

2. Polynomials: • p = [1 -2 3 6] , y = polyval(p, x)%definition • Examples:Poly_01.m , Poly_02.m • c = conv(a,b) % multiplication • Example:Poly_03.m • [q, r]=deconv(a,b)% division • Example: Poly_04.m • c = polyder(p) %derivative • Example:Poly_05.m • c = polyder(a,b) %derivative of product • Example: Poly_06.m • [n,d] = polyder(a,b) %derivative of division • Example: Poly_07.m Mathematical Modeling and Simulation 2

3. ……..Polynomials: • intgrl = polyint(p) integral of polynomial ‘p’ • Example: Poly_09.m • intgrl = polyint(p, c) integral of polynomial ‘p’ • Example: Poly_10.m c a constant of integration • r = roots(p)roots of polynomial ‘p’ • Example: Poly_11.m • p = poly(r) polynomial of roots ‘r’ • Example: Poly_12.m • p = poly(x)x must be a square matrix • Example: Poly_13.mp is characteristic polynomial Mathematical Modeling and Simulation 3

4. Interpolation I: • interp1(x,y,a), Example:InterpFit_01.m • interp1(x,y,a,’linear’), InterpFit_01b.m • interp1(x,y,a,’cubic’), • interp1(x,y,a,’spline’), • Interp1(x,y,a,’nearest’) • interp2(x,y,z,a,b,’ …….. ‘) , [xx,yy]=meshgrid(x,y), mesh() • Example: InterpFit_02.m • interp3 • interp1q, %it is quicker than ‘interp1’ on non-uniformly • spaced data because it does no input checking • interpft, • interpn Mathematical Modeling and Simulation 4

5. Interpolation II: • tri=delaunay(x,y), trimesh(tri,x,y,z), • tsearch(x,y,tri,[x b],[c d]), dsearch • Example:RandomDataInterp_01 • [pts,area] = convhull(x,y) Example: RandomDataInterp_02 • voronoi(x,y) Example:RandomDataInterp_03 • griddata Example:RandomDataInterp_04 Mathematical Modeling and Simulation 5

6. Curve Fitting: • p = polyfit(x,y,n) Example: PolyFits_01.m • [p, s] = polyfit(x,y,n) • [p,s,μ ] = polyfit(x,y,n) • yi = spline(x,y,xi) Example: SplineFits_01.m • pp=spline(x,y), yi=ppval(pp,xi) • hp = pchip(x,y), Example: HermiteSplineFits_01.m Mathematical Modeling and Simulation 6

7. Colormap Mathematical Modeling and Simulation 7

8. Test Matrices: binomial cauchy chebspec chebvand chow circul clement compar condex cycol dorr dramadah fiedler forsythe frank gearmat gcdmat grcar hanowa house invhess invol ipjfact jordbloc kahan kms krylov lauchli lehmer leslie lesp lotkin minij moler neumann orthog parter pei poisson prolate randcolu randcorr randhess randjorth rando randsvd redheff riemann ris smoke toeppd tridiag triw wathen wilk A=gallery(‘binomial’, n) Mathematical Modeling and Simulation 8

9. End Mathematical Modeling and Simulation 9