Problem 2

README.md

@@ -1 +1,62 @@
-# 05_curve_fitting
+# 05_curve_fitting
+ME 3255
+Homework 5
+
+##Problem 2
+Least Squares function
+```matlab
+function [a,fx,r2] = least_squares(Z,y);
+  % inputs: Z matrix and measurements y
+  % outputs:
+  % a   - best fit constant
+  % fx  - function at xi
+  % r2  - coefficient of determination
+  a = Z/y;
+  fx = Z*a;
+  e = y-fx;
+  st = std(y);
+  sr = std(e);
+  r2 = (st-sr)/st;
+end
+```
+set defaults
+% Part A
+xa = [1 2 3 4 5]';
+ya = [2.2 2.8 3.6 4.5 5.5]';
+Z = [xa.^0,xa.^1,1./xa];
+[a,fx,r2] = least_squares(Z,ya)
+plot(xa,fx,xa,ya,'o','Linewidth',2)
+
+% Part B
+xb = [0 2 4 6 8 10 12 14 16 18]';
+yb = [21.5 20.84 23.19 22.69 30.27 40.11 43.31 54.79 70.88 89.48]';
+Z = [xb.^0,-xb.^1,xb.^2];
+[a,fx,r2] = least_squares(Z,yb)
+plot(xb,fx,xb,yb,'o','Linewidth',2)
+
+% Part C
+xc = [0.5 1 2 3 4 5 6 7 9]';
+yc = [6 4.4 3.2 2.7 2.2 1.9 1.7 1.4 1.1]';
+Z = [exp(-1.5*xc),exp(-0.3*xc),exp(-0.05*xc)];
+[a,fx,r2] = least_squares(Z,yc)
+plot(xc,fx,xc,yc,'o','Linewidth',2)
+
+% Part D
+xd = [0.00000000e+00 1.26933037e-01 2.53866073e-01 3.80799110e-01 5.07732146e-01 6.34665183e-01 7.61598219e-01 8.88531256e-01 1.01546429e+00 1.14239733e+00 1.26933037e+00 1.39626340e+00 1.52319644e+00 1.65012947e+00 1.77706251e+00 1.90399555e+00 2.03092858e+00 2.15786162e+00 2.28479466e+00 2.41172769e+00 2.53866073e+00 2.66559377e+00 2.79252680e+00 2.91945984e+00 3.04639288e+00 3.17332591e+00 3.30025895e+00 3.42719199e+00 3.55412502e+00 3.68105806e+00 3.80799110e+00 3.93492413e+00 4.06185717e+00 4.18879020e+00 4.31572324e+00 4.44265628e+00 4.56958931e+00 4.69652235e+00 4.82345539e+00 4.95038842e+00 5.07732146e+00 5.20425450e+00 5.33118753e+00 5.45812057e+00 5.58505361e+00 5.71198664e+00 5.83891968e+00 5.96585272e+00 6.09278575e+00 6.21971879e+00 6.34665183e+00 6.47358486e+00 6.60051790e+00 6.72745093e+00 6.85438397e+00 6.98131701e+00 7.10825004e+00 7.23518308e+00 7.36211612e+00 7.48904915e+00 7.61598219e+00 7.74291523e+00 7.86984826e+00 7.99678130e+00 8.12371434e+00 8.25064737e+00 8.37758041e+00 8.50451345e+00 8.63144648e+00 8.75837952e+00 8.88531256e+00 9.01224559e+00 9.13917863e+00 9.26611167e+00 9.39304470e+00 9.51997774e+00 9.64691077e+00 9.77384381e+00 9.90077685e+00 1.00277099e+01 1.01546429e+01 1.02815760e+01 1.04085090e+01 1.05354420e+01 1.06623751e+01 1.07893081e+01 1.09162411e+01 1.10431742e+01 1.11701072e+01 1.12970402e+01 1.14239733e+01 1.15509063e+01 1.16778394e+01 1.18047724e+01 1.19317054e+01 1.20586385e+01 1.21855715e+01 1.23125045e+01 1.24394376e+01 1.25663706e+01]';
+yd=[9.15756288e-02 3.39393873e-01 6.28875306e-01 7.67713096e-01 1.05094584e+00 9.70887288e-01 9.84265740e-01 1.02589034e+00 8.53218113e-01 6.90197665e-01 5.51277193e-01 5.01564914e-01 5.25455797e-01 5.87052838e-01 5.41394658e-01 7.12365594e-01 8.14839678e-01 9.80181855e-01 9.44430709e-01 1.06728057e+00 1.15166322e+00 8.99464065e-01 7.77225453e-01 5.92618124e-01 3.08822183e-01 -1.07884730e-03 -3.46563271e-01 -5.64836023e-01 -8.11931510e-01 -1.05925186e+00 -1.13323611e+00 -1.11986890e+00 -8.88336727e-01 -9.54113139e-01 -6.81378679e-01 -6.02369117e-01 -4.78684439e-01 -5.88160325e-01 -4.93580777e-01 -5.68747320e-01 -7.51641934e-01 -8.14672884e-01 -9.53191554e-01 -9.55337518e-01 -9.85995556e-01 -9.63373597e-01 -1.01511061e+00 -7.56467517e-01 -4.17379564e-01 -1.22340361e-01 2.16273929e-01 5.16909714e-01 7.77031694e-01 1.00653798e+00 9.35718089e-01 1.00660116e+00 1.11177057e+00 9.85485116e-01 8.54344900e-01 6.26444042e-01 6.28124048e-01 4.27764254e-01 5.93991751e-01 4.79248018e-01 7.17522492e-01 7.35927848e-01 9.08802925e-01 9.38646871e-01 1.13125860e+00 1.07247935e+00 1.05198782e+00 9.41647332e-01 6.98801244e-01 4.03193543e-01 1.37009682e-01 -1.43203880e-01 -4.64369445e-01 -6.94978252e-01 -1.03483196e+00 -1.10261288e+00 -1.12892727e+00 -1.03902484e+00 -8.53573083e-01 -7.01815315e-01 -6.84745997e-01 -6.14189417e-01 -4.70090797e-01 -5.95052432e-01 -5.96497000e-01 -5.66861911e-01 -7.18239679e-01 -9.52873043e-01 -9.37512847e-01 -1.15782985e+00 -1.03858206e+00 -1.03182712e+00 -8.45121554e-01 -5.61821980e-01 -2.83427014e-01 -8.27056140e-02]';
+Z = [sin(xd),sin(3*xd)];
+[a,fx,r2] = least_squares(Z,yd)
+plot(xd,fx,xd,yd,'o','Linewidth',2)
+```
+Output:
+Part A: 0.9801
+![Part A](./figure1.png)
+
+Part B: 0.9112
+![Part B](./figure2.png)
+
+Part C: 0.9462
+![Part C](./figure3.png)
+
+Part D: 0.9219  
+![Part D](./figure4.png)

data.csv

@@ -0,0 +1,26 @@
+Data
+Re(e-4) C_D
+2 0.52
+5.8 0.52
+16.8 0.52
+27.2 0.5
+29.9 0.49
+33.9 0.44
+36.3 0.18
+40 0.074
+46 0.067
+60 0.08 
+100 0.12
+200 0.16
+400 0.19
+
+mu=1.78e-5;
+V=linspace(4,40,1000);
+D=23.5;
+rho=1.3;
+Re=rho*V*D/mu*1e-4;
+Cd_int = interp1(data(:,1), (data(:,2), Re, 'spline');
+Cd_int_pc  = interp1(data(:,1), (data(:,2), Re, 'pchip');
+
+plot(Re, Cd_int, Re, Cd_int_pc, data(:,1),data(:,2),'o','MarkerSize',10,LineWidth
+legend('cubic','pchip','data')

least_squares.m

@@ -0,0 +1,13 @@
+function [a,fx,r2] = least_squares(Z,y);
+  % inputs: Z matrix and measurements y
+  % outputs:
+  % a   - best fit constant
+  % fx  - function at xi
+  % r2  - coefficient of determination
+  a = Z/y;
+  fx = Z*a;
+  e = y-fx;
+  st = std(y);
+  sr = std(e);
+  r2 = (st-sr)/st;
+end

problem_2_data.m

@@ -0,0 +1,19 @@
+
+% part a
+xa=[1 2 3 4 5]';
+ya=[2.2 2.8 3.6 4.5 5.5]';
+
+% part b
+
+xb=[0 2 4 6 8 10 12 14 16 18]'
+yb=[21.5 20.84 23.19 22.69 30.27 40.11 43.31 54.79 70.88 89.48]';
+
+% part c
+
+xc=[0.5 1 2 3 4 5 6 7 9]';
+yc=[6 4.4 3.2 2.7 2.2 1.9 1.7 1.4 1.1]';
+
+% part d
+
+xd=[0.00000000e+00 1.26933037e-01 2.53866073e-01 3.80799110e-01 5.07732146e-01 6.34665183e-01 7.61598219e-01 8.88531256e-01 1.01546429e+00 1.14239733e+00 1.26933037e+00 1.39626340e+00 1.52319644e+00 1.65012947e+00 1.77706251e+00 1.90399555e+00 2.03092858e+00 2.15786162e+00 2.28479466e+00 2.41172769e+00 2.53866073e+00 2.66559377e+00 2.79252680e+00 2.91945984e+00 3.04639288e+00 3.17332591e+00 3.30025895e+00 3.42719199e+00 3.55412502e+00 3.68105806e+00 3.80799110e+00 3.93492413e+00 4.06185717e+00 4.18879020e+00 4.31572324e+00 4.44265628e+00 4.56958931e+00 4.69652235e+00 4.82345539e+00 4.95038842e+00 5.07732146e+00 5.20425450e+00 5.33118753e+00 5.45812057e+00 5.58505361e+00 5.71198664e+00 5.83891968e+00 5.96585272e+00 6.09278575e+00 6.21971879e+00 6.34665183e+00 6.47358486e+00 6.60051790e+00 6.72745093e+00 6.85438397e+00 6.98131701e+00 7.10825004e+00 7.23518308e+00 7.36211612e+00 7.48904915e+00 7.61598219e+00 7.74291523e+00 7.86984826e+00 7.99678130e+00 8.12371434e+00 8.25064737e+00 8.37758041e+00 8.50451345e+00 8.63144648e+00 8.75837952e+00 8.88531256e+00 9.01224559e+00 9.13917863e+00 9.26611167e+00 9.39304470e+00 9.51997774e+00 9.64691077e+00 9.77384381e+00 9.90077685e+00 1.00277099e+01 1.01546429e+01 1.02815760e+01 1.04085090e+01 1.05354420e+01 1.06623751e+01 1.07893081e+01 1.09162411e+01 1.10431742e+01 1.11701072e+01 1.12970402e+01 1.14239733e+01 1.15509063e+01 1.16778394e+01 1.18047724e+01 1.19317054e+01 1.20586385e+01 1.21855715e+01 1.23125045e+01 1.24394376e+01 1.25663706e+01]';
+yd=[9.15756288e-02 3.39393873e-01 6.28875306e-01 7.67713096e-01 1.05094584e+00 9.70887288e-01 9.84265740e-01 1.02589034e+00 8.53218113e-01 6.90197665e-01 5.51277193e-01 5.01564914e-01 5.25455797e-01 5.87052838e-01 5.41394658e-01 7.12365594e-01 8.14839678e-01 9.80181855e-01 9.44430709e-01 1.06728057e+00 1.15166322e+00 8.99464065e-01 7.77225453e-01 5.92618124e-01 3.08822183e-01 -1.07884730e-03 -3.46563271e-01 -5.64836023e-01 -8.11931510e-01 -1.05925186e+00 -1.13323611e+00 -1.11986890e+00 -8.88336727e-01 -9.54113139e-01 -6.81378679e-01 -6.02369117e-01 -4.78684439e-01 -5.88160325e-01 -4.93580777e-01 -5.68747320e-01 -7.51641934e-01 -8.14672884e-01 -9.53191554e-01 -9.55337518e-01 -9.85995556e-01 -9.63373597e-01 -1.01511061e+00 -7.56467517e-01 -4.17379564e-01 -1.22340361e-01 2.16273929e-01 5.16909714e-01 7.77031694e-01 1.00653798e+00 9.35718089e-01 1.00660116e+00 1.11177057e+00 9.85485116e-01 8.54344900e-01 6.26444042e-01 6.28124048e-01 4.27764254e-01 5.93991751e-01 4.79248018e-01 7.17522492e-01 7.35927848e-01 9.08802925e-01 9.38646871e-01 1.13125860e+00 1.07247935e+00 1.05198782e+00 9.41647332e-01 6.98801244e-01 4.03193543e-01 1.37009682e-01 -1.43203880e-01 -4.64369445e-01 -6.94978252e-01 -1.03483196e+00 -1.10261288e+00 -1.12892727e+00 -1.03902484e+00 -8.53573083e-01 -7.01815315e-01 -6.84745997e-01 -6.14189417e-01 -4.70090797e-01 -5.95052432e-01 -5.96497000e-01 -5.66861911e-01 -7.18239679e-01 -9.52873043e-01 -9.37512847e-01 -1.15782985e+00 -1.03858206e+00 -1.03182712e+00 -8.45121554e-01 -5.61821980e-01 -2.83427014e-01 -8.27056140e-02]';

setdefaults.m

@@ -0,0 +1,3 @@
+set(0, 'defaultAxesFontSize', 16)
+set(0,'defaultTextFontSize',14)
+set(0,'defaultLineLineWidth',3)

sphere_drag.m

@@ -0,0 +1,10 @@
+function [Cd_out] = sphere_drag(Re_in,spline_type)
+    % interpolation for drag coeff based upon Reynolds number
+    % output is 
+    % Cd: drag coeff
+    % input is 
+    % Re_in: Reynolds number
+    % and 
+    % spline_type: 'linear', 'cubic', or 'spline'/'pchip'
+
+