# 481 - Hermite

## 352 days ago by Professor481

# This cell in Sage mode var('u') H0 = (1-u)^2*(1+2*u) H1 = (1-u)^2*u H2 = (u-1)*u^2 H3 = (3-2*u)*u^2 G = plot(H0,(u,0,1),color='black')+plot(H1,(u,0,1),color='red')+plot(H2,(u,0,1),color='green')+plot(H3,(u,0,1),color='blue') show(G,aspect_ratio=1)

The rest of this is in octave mode

function y = H0(u) y = (1-u) .^2 .* (1+2*u) end function y = H1(u) y = (1-u) .^2 .*u end function y = H2(u) y = (u-1) .* u .^2 end function y = H3(u) y = (3-2 .*u) .* u .^2 end n = 3 x = [-1 2 3 7] y = [5 4 2 3] m = [1 -2 1 -1] for k = 2:n+1 xx = x(k-1):0.5:x(k); delx = x(k)-x(k-1); yy = H0( (xx-x(k-1))/(x(k)-x(k-1)) )*y(k-1) + H1( (xx-x(k-1))/(x(k)-x(k-1)) )*m(k-1)*delx + H2( (xx-x(k-1))/(x(k)-x(k-1)) )*m(k)*delx + H3( (xx-x(k-1))/(x(k)-x(k-1)) )*y(k); plot(xx,yy) hold on end
 n = 3 x = -1 2 3 7 y = 5 4 2 3 m = 1 -2 1 -1 y = 1 0.925926 0.740741 0.5 0.259259 0.0740741 0 y = 0 0.115741 0.148148 0.125 0.0740741 0.0231481 0 y = -0 -0.0231481 -0.0740741 -0.125 -0.148148 -0.115741 0 y = 0 0.0740741 0.259259 0.5 0.740741 0.925926 1 y = 1 0.5 0 y = 0 0.125 0 y = -0 -0.125 0 y = 0 0.5 1 y = 1 0.957031 0.84375 0.683594 0.5 0.316406 0.15625 0.0429688 0 y = 0 0.0957031 0.140625 0.146484 0.125 0.0878906 0.046875 0.0136719 0 y = -0 -0.0136719 -0.046875 -0.0878906 -0.125 -0.146484 -0.140625 -0.0957031 0 y = 0 0.0429688 0.15625 0.316406 0.5 0.683594 0.84375 0.957031 1 6 |-----------------------------------------------------------------------\ | | + + + + | | ********* | | *** * | | ** * | 5 |-+ * * +-| | * | | * | | * | | * | 4 |-+ # +-| | # | | # $$| | #$$ $$| | #$$ | 3 |-+ # $$+-| | #  | | ##  | | ##$$ | | + + $$+ + | 2 |-----------------------------------------------------------------------\ | -2 0 2 4 6 8  n = 3 x = -1 2 3 7 y = 5 4 2 3 m = 1 -2 1 -1 y = 1 0.925926 0.740741 0.5 0.259259 0.0740741 0 y = 0 0.115741 0.148148 0.125 0.0740741 0.0231481 0 y = -0 -0.0231481 -0.0740741 -0.125 -0.148148 -0.115741 0 y = 0 0.0740741 0.259259 0.5 0.740741 0.925926 1 y = 1 0.5 0 y = 0 0.125 0 y = -0 -0.125 0 y = 0 0.5 1 y = 1 0.957031 0.84375 0.683594 0.5 0.316406 0.15625 0.0429688 0 y = 0 0.0957031 0.140625 0.146484 0.125 0.0878906 0.046875 0.0136719 0 y = -0 -0.0136719 -0.046875 -0.0878906 -0.125 -0.146484 -0.140625 -0.0957031 0 y = 0 0.0429688 0.15625 0.316406 0.5 0.683594 0.84375 0.957031 1 6 |-----------------------------------------------------------------------| | + + + + | | ********* | | *** * | | ** * | 5 |-+ * * +-| | * | | * | | * | | * | 4 |-+ # +-| | # | | #$$$| | # $ | | # $$| 3 |-+ #$$ $+-| | #$ | | ## \$ | | ## $$| | + +$$ + + | 2 |-----------------------------------------------------------------------| -2 0 2 4 6 8