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

𝐴𝑐

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 k1 = 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 x1 = 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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