added splines and integrals

16_splines/Newtint.m

@@ -0,0 +1,34 @@
+function yint = Newtint(x,y,xx)
+% Newtint: Newton interpolating polynomial
+% yint = Newtint(x,y,xx): Uses an (n - 1)-order Newton
+%   interpolating polynomial based on n data points (x, y)
+%   to determine a value of the dependent variable (yint)
+%   at a given value of the independent variable, xx.
+% input:
+%   x = independent variable
+%   y = dependent variable
+%   xx = value of independent variable at which
+%        interpolation is calculated
+% output:
+%   yint = interpolated value of dependent variable
+
+% compute the finite divided differences in the form of a
+% difference table
+n = length(x);
+if length(y)~=n, error('x and y must be same length'); end
+b = zeros(n,n);
+% assign dependent variables to the first column of b.
+b(:,1) = y(:); % the (:) ensures that y is a column vector.
+for j = 2:n
+  for i = 1:n-j+1
+    b(i,j) = (b(i+1,j-1)-b(i,j-1))/(x(i+j-1)-x(i));
+  end
+end
+%b
+% use the finite divided differences to interpolate
+xt = 1;
+yint = b(1,1);
+for j = 1:n-1
+  xt = xt*(xx-x(j));
+  yint = yint+b(1,j+1)*xt;
+end

16_splines/simpson3.m

@@ -0,0 +1,23 @@
+function I = simpson3(func,a,b,n,varargin)
+% simpson3: composite simpson's 1/3 rule
+%   I = simpson3(func,a,b,n,pl,p2,...):
+%       composite trapezoidal rule
+% input:
+%   func = name of function to be integrated
+%   a, b = integration limits
+%   n = number of segments (default = 100)
+%   pl,p2,... = additional parameters used by func
+% output:
+%   I = integral estimate
+if nargin<3,error('at least 3 input arguments required'),end
+if ~(b>a),error('upper bound must be greater than lower'),end
+if nargin<4|isempty(n),n=100;end
+x = a; h = (b - a)/n;
+
+xvals=linspace(a,b,n+1);
+fvals=func(xvals,varargin{:});
+s=fvals(1);
+s = s + 4*sum(fvals(2:2:end-1));
+s = s + 2*sum(fvals(3:2:end-2));
+s = s + fvals(end);
+I = (b - a) * s/(3*n);

16_splines/trap.m

@@ -0,0 +1,22 @@
+function I = trap(func,a,b,n,varargin)
+% trap: composite trapezoidal rule quadrature
+%   I = trap(func,a,b,n,pl,p2,...):
+%       composite trapezoidal rule
+% input:
+%   func = name of function to be integrated
+%   a, b = integration limits
+%   n = number of segments (default = 100)
+%   pl,p2,... = additional parameters used by func
+% output:
+%   I = integral estimate
+if nargin<3,error('at least 3 input arguments required'),end
+if ~(b>a),error('upper bound must be greater than lower'),end
+if nargin<4|isempty(n),n=100;end
+
+x = a; h = (b - a)/n;
+xvals=linspace(a,b,n);
+fvals=func(xvals,varargin{:});
+s=func(a,varargin{:});
+s = s + 2*sum(fvals(2:n-1));
+s = s + func(b,varargin{:});
+I = (b - a) * s/(2*n);

17_integrals_and_derivatives/.ipynb_checkpoints/lecture_20-checkpoint.ipynb

@@ -0,0 +1,6 @@
+{
+ "cells": [],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

17_integrals_and_derivatives/stainless_steel_psi.jpg.dat

@@ -0,0 +1,12 @@
+1.0208494318e-05  1.6556901722
+0.00241601032192  33.1999376148
+0.00420249682757  53.9506164087
+0.00603492155765  82.1777412288
+0.00844582763241  114.552704897
+0.00959768607462  122.017666367
+0.0207793901842  141.840010208
+0.0369377352739  161.610673548
+0.0574942399989  177.181817537
+0.0774314294019  181.959392878
+0.100609815751  174.241771174
+0.117644389936  156.618719826

17_integrals_and_derivatives/structural_steel_psi.jpg.dat

@@ -0,0 +1,13 @@
+1.0208494318e-05  1.6556901722
+0.00180179924712  23.2370851913
+0.00242111456908  34.0306538399
+0.00298938741945  36.5170602372
+0.00410551613155  38.1670081313
+0.0113042060414  39.7537909669
+0.026807506079  42.9158720819
+0.0450807109082  46.8799580317
+0.063896667352  49.1768692533
+0.0937667217264  50.5282186886
+0.134122601181  48.4475999405
+0.194912483429  42.0009357786
+0.224198952211  38.3737301413