es117 - introduction to scientific computing with...

LECTURE 12: APPLICATIONS IN SCIENCE

AND ENGINEERING

ES117 - Introduction to Scientific Computing

with Matlab

MATLAB Applications

Polynomials

Curve fitting

Linear equations

Interpolation

Polynomials

Polynomial anxn + an-1 xn-1 ... + a1x + a0

(n is the degree of the polynomial)

A polynomial in MATLAB is represented by a row

vector in which the elements are the coefficients.

Example: polynomial y=4x2 + 10x + 3

x=1 y=4+10+3=17

x=2 y=16+20+3=39

x=3 y=36+30+3=69 .......

Polynomials

A polynomial in MATLAB is represented by a row

vector in which the elements are the coefficients.

Example: To define the polynomial y=4x2 + 10x + 3

a = [4 10 3]

a =

4 10 3

Polynomials

To evaluate a polynomial at point x: y= polyval(a,x)

Example: To evaluate the polynomial y=4x2 + 10x + 3 at x=2 a = [4 10 3]

y=polyval(a,2)

y =

39

Polynomials

To find the roots of a polynomial (values for which the polynomial is equal to 0) r=roots(a)

Example: To find the roots of the polynomial 4x2 + 10x + 3 a = [4 10 3]

r=roots(a)

r =

-2.1514

-0.3486

Polynomials

To plot a polynomial x = d:e:f;

y = polyval(a,x);

plot(x,y)

Example: To plot the polynomial 4x2 + 10x + 3 in the interval [0 25] in 1 unit increments a = [4 10 3];

x = 0:25;

y = polyval(a,x);

plot(x,y)

Polynomials

a = [4 10 3];

x = 0:25;

y = polyval(a,x);

plot(x,y)

Curve fitting – Basic fitting

To find an algebraic relationship that best fits a given

set of data

To explore the best possible

fit:

Basic Fitting in the Figure

window’s Tools menu lets

you fit a polynomial curve

(upto 10th order) to your

data.

Curve fitting

Straight line (linear fit)

Step 1: Plot raw data

x=[5 10 20 50 100]

y=[15 33 53 140 301]

plot(x,y,'o')

Curve fitting

Step 2: Use built in Basic Fitting to do a linear fit

Figure window – Tools – Basic Fitting

Curve fitting

Step 3: Fit a linear curve and display the equation

Curve fitting

Comparing different fits

Step 1: Plot raw data

x=0:pi/30:pi/3;

y = sin(x) + rand(size(x))/100;

plot(x,y,'o')

Curve fitting

Step 2: Use Basic Fitting to do a quadratic and cubic

fit.

Curve fitting

Norm of residuals

for the cubic fit is

lower than that

for quadratic fit

cubic fit should be

chosen.

Curve fitting – Polynomial functions

Given two vectors x and y, the command

a=polyfit(x,y,n) fits a polynomial of order n through

the data points (xi,yi) and returns n+1 coefficients (in

the decreasing order) for the powers of x in the row

vector a.


x=0:5;

y=[0,20,60,68,77,110];

coef=polyfit(x,y,1) first order Output: What is the equation that describes the best fit linear model for this data set? y = 20.8286 x + 3.7619


How can you calculate the y values predicted by this

best fit polynomial for the given x values?

(20.8286x) + 3.7619 a=[20.826 3.7619] y=polyval(a,2) y=20.826x2+3.7619 y= 45.4190

Matrix multiplication

Using (.) means ‘do the operation elementwise’ a*b is matrix multiplication. a.*b is elementwise multiplication. >> a=[2 3;1 5]; b=[0 3;2 0]; >> a*b ans = 6 6 10 3 >> a.*b ans =0 9 2 0

Solving linear equations

Solution of a linear systems of equations

3x + 2y − z = 10

−x + 3y + 2z = 5

x − y − z = −1

If we define the matrix A as

And the vectors x and b as


3x + 2y − z = 10

−x + 3y + 2z = 5

x − y − z = −1

A x = b

The solution may then be written as

where A-1 is the matrix inverse of A (the matrix which

gives the identity matrix when multiplied by A)


Using matrix division

3x + 2y − z = 10

−x + 3y + 2z = 5

x − y − z = −1

A x = b

x=-2

y=5

z=-6


Using matrix division

Interpolation

In some applications we want to estimate a variable’s

value between the data points. This process is called

interpolation.

Suppose we have the following temperature

measurements. The measurements at 8 and 10 A.M.

are missing for some reason, perhaps because of

equipment malfunction.

Interpolation

53 ???

64 ???

Interpolation

Linear interpolation in MATLAB is obtained with the interp1 and

interp2 functions.

For example, the following session produces an estimate of the

temperatures at 8 and 10 A.M. from the preceding data. The

vectors x and y contain the times and temperatures, respectively.

>>x = [7, 9, 11, 12];

>>y = [49, 57, 71, 75];

>>x_int = [8, 10];

>>interp1(x,y,x_int)

ans =

53

64

Interpolation

The values of the independent variable in the vector x (time values in the example) must be in ascending order.

The values in the interpolation vector x_int must lie within the range of the values in x. Thus, we cannot use the interp1 function to estimate the temperature at 6 A.M., for example.

Interpolation

The interp1 function can be used to interpolate in a

table of values by defining y to be a matrix instead of a

vector.

For example, suppose that we now have temperature

measurements at three locations.

Interpolation

We define x as before, but now we de ne y to be a matrix whose three

columns contain the second, third, and fourth columns of the

preceding table. The following session produces an estimate of the

temperatures at 8 and 10 A.M. at each location.

>>x = [7, 9, 11, 12]’;

>>y(:,1) = [49, 57, 71, 75]’;

>>y(:,2) = [52, 60, 73, 79]’;

>>y(:,3) = [54, 61, 75, 81]’;

>>x_int = [8, 10]’;

>>interp1(x,y,x_int)

ans =

53.0000 56.0000 57.5000

64.0000 65.5000 68.0000

The estimated temperatures at 8 A.M. at each location are 53, 56, and 57.5, respectively. At 10 A.M. the estimated temperatures are 64, 65.5, and 68.

Two-Dimensional Interpolation

Now suppose that we have

temperature measurements

at four locations at 7 A.M. These

locations are at the corners of a

rectangle 1 mile wide and 2 mile

long.

The temperature is a function of

two variables, the coordinates x

and y.


MATLAB provides the interp2 function to

interpolate functions of two variables.

If the function is written as z f(x,y) and we wish to

estimate the value of z for x=xi and y=yi, the syntax is:

interp2(x,y,z,x_i,y_i)


Suppose we want to estimate

the temperature at the point

whose coordinates are (0.6,

1.5). Put the x coordinates in

the vector x and the y

coordinates in the vector y.

Then put the temperature

measurements in a matrix z

such that going across a row

represents an increase in x and

going down a column

represents an increase in

y.

>>x = [0,1];

>>y = [0,2];

>>z = [49,54;53,57]

z =

49 54

53 57

>>interp2(x,y,z,0.6,1.5)

ans =

54.5500

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