Computational Method

For each Curve Fitting Problem, please also draw the graph that contains the data set and the
curve fitting.
Problem 1 (interpolation)
The following data come from a table that was measured with high precision. Use the best
polynomial interpolation numerical method to determine y at x = 3.5.
Remark: Do not use built in functions.

x 0 1.8 5 6 8.2 9.2 12
y 26 16.145 5.375 3.5 2.015 2.54 8

Problem 2 (Curve fitting)
The mast of a sailboat has a cross-sectional area of 10.65 (cm2) and is constructed of an
experimental aluminum alloy. Tests were performed to define the relationship between stress and
strain. The test results are:

Strain 0.0032 0.0045 0.0055 0.0016 0.0085 0.0005
Stress (N/cm2) 4970 5170 5500 3590 6900 1240

The stress caused by wind can be computed as 𝐹
𝐴
𝑐
where F = force in the mast and
Ac = mast’s
cross-sectional area. This value can then be substituted into Hooke’s law to determine the mast’s
deflection,
∆𝐿 = 𝜀𝐿, where L = the mast’s length. If the wind force is 25,000 (N), curve fit the
data and then use it to estimate the deflection of a 9-m mast.
REMARK: Use matrix method. You may possibly want to fix the data with polynomials with
increasing order until the results become reliable.

Homework 3
Problem 3 (Curve fitting)
Andrade’s equation has been proposed as a model of the effect of temperature on viscosity:
 = DeB T / a
where = dynamic viscosity of water (10 / 3 2 N s m ), Ta = absolute temperature (K), and D and
B are parameters. Fit this model to the following data for water:

T 0 5 10 20 30 40
μ 1.787 1.519 1.307 1.002 0.7975 0.6529

REMARK: Use built-in functions
Problem 4 (Least-Square fitting)
Three disease-carrying organisms decay exponentially in seawater according to the following
model:
p t Ae Be Ce ( ) = + + – – – 1.5 0.3 0.05 t t t
Use general linear least-squares to estimate the initial concentration of each organism (A, B,
and C) given the following measurements:

t 0.5 1 2 3 4 5 6 7 9
p(t) 6 4.4 3.2 2.7 2 1.9 1.7 1.4 1.1

(a) Use build-in function.
(b) Apply matrix method.
Problem 5 (General)
Enzymatic reactions are used extensively to characterize biologically mediated reactions. The
following is an example of a model that is used to fit such reactions:
3
0 3
k S m
v
K S
=
+

Homework 3
where
v0 = the initial rate of the reaction (M/s), [S] = the substrate concentration (M), and km
and K are parameters. The following data can be fit with this model:

[S] (M) v0 (M/s)
0.01
0.05
0.1
0.5
1
5
10
50
100

6.078 10 11
7.595 10 9
6.063 10 8
5.788 10 6
1.737 10 5
2.423 10 5
2.430 10 5
2.431 10 5
2.431 10 5
(a) Use a transformation to linearize the model and evaluate the parameters. Display the data
and the model fit on a graph.
(b) Perform the same evaluation as in (a) but use nonlinear regression.
Problem 6 (Eigenvalues)
Given the 4 mass 5 spring system shown below:
Assume k
1 = 30 N/m, k2 = 25 N/m, k3 = 25 N/m, m1 = 1 kg, m2 = 0.75 kg, and m3 = 0.5 kg
(a) Use the power method to find the max eigenvalue and the eigenvector with an error of
1e-6%
(b) Use the built in Matlab function to find the eigenvalues and eigenvector
Problem 7 (1D Optimization)
Homework 3
Given the function:
𝑓(𝑥) = 4𝑥 – 1.8𝑥2 + 1.2𝑥3 – 0.3 ∗ 𝑥4
(a) Use the golden search method with the upper and lower bounds of -2 and 4 with an error
of 1e-6%
(b) Use Parabolic Interpolation for 5 iterations with x
1 = 1.75, x2 = 2, and x3 = 2.5
(c) Use built in function using the upper and lower limits from part a. Have the results of
each iteration displayed.
Problem 8 (Multidimensional Optimization)
A finite-element model of a cantilever beam subject to loading and moments (Figure above) is
given by optimizing:
𝑓(𝑥, 𝑦) = 5𝑥2 – 5𝑥𝑦 + 2.5𝑦2 – 𝑥 – 1.5𝑦
where x = end displacement and y = end moment. Find the values of x and y that minimize f(x,
y).
Remarks: Have fminsearch display the results of each iteration.

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