ch 7 Applications
APPLICATIONS
Polynomials
The coefficients of a polynomial can be represented as a row vector and using the
linspace command the polynomial can be evaluated for each element of the row
vector with the command polyval. For example the polynomial
y =x4 -2x3 +4x2 -8x+10
can be evaluated at 6 points from x = 0 to x = 10 with the statements
P = [1 –2 4 –8 10];
x = linspace(0,10,6);
polyval(P,x)
ans
10
10
170
970
3274
8330
The roots of a polynomial can be found using the roots command. Consider the
poynomial
(x – 3)(x + 2) = x2 – x – 6 which has the roots x = 3 and x – 2.
The Matlab commands to evaluate the roots are
P = [1 –1 –6]
roots(P)

3
2
Another example in which the roots are not real is for the polynomial
y =x4 -2x3 +4x2 -8x+10 . The roots are complex where i  1 .
Matlab statements
P = [1 –2 4 –8 10]
-0.4199 + 1.8588i
roots(P)

-0.4199 - 1.8588i
1.4199 + 0.8588i
1.4199 - 0.8588i
Move to later chapter
POLYVAL Evaluate polynomial.
Y = POLYVAL(P,X), when P is a vector of length N+1 whose elements
are the coefficients of a polynomial, is the value of the
polynomial evaluated at X.
Y = P(1)*X^N + P(2)*X^(N-1) + ... + P(N)*X + P(N+1)
If X is a matrix or vector, the polynomial is evaluated at all
points in X. See also POLYVALM for evaluation in a matrix sense.
Y = POLYVAL(P,X,[],MU) uses XHAT = (X-MU(1))/MU(2) in place of X.
The centering and scaling parameters MU are optional output
computed by POLYFIT.
[Y,DELTA] = POLYVAL(P,X,S) or [Y,DELTA] = POLYVAL(P,X,S,MU) uses
the optional output structure S provided by POLYFIT to generate
error estimates, Y +/- delta. If the errors in the data input to
POLYFIT are independent normal with constant variance, Y +/- DELTA
contains at least 50% of the predictions.
See also POLYFIT, POLYVALM.
POLYFIT Fit polynomial to data.
POLYFIT(X,Y,N) finds the coefficients of a polynomial P(X) of
degree N that fits the data, P(X(I))~=Y(I), in a least-squares sense.
[P,S] = POLYFIT(X,Y,N) returns the polynomial coefficients P and a
structure S for use with POLYVAL to obtain error estimates on
predictions. If the errors in the data, Y, are independent normal
with constant variance, POLYVAL will produce error bounds which
contain at least 50% of the predictions.
The structure S contains the Cholesky factor of the Vandermonde
matrix (R), the degrees of freedom (df), and the norm of the
residuals (normr) as fields.
[P,S,MU] = POLYFIT(X,Y,N) finds the coefficients of a polynomial
in XHAT = (X-MU(1))/MU(2) where MU(1) = mean(X) and MU(2) = std(X).
This centering and scaling transformation improves the numerical
properties of both the polynomial and the fitting algorithm.
Warning messages result if N is >= length(X), if X has repeated, or
nearly repeated, points, or if X might need centering and scaling.
See also POLY, POLYVAL, ROOTS.
ROOTS Find polynomial roots.
ROOTS(C) computes the roots of the polynomial whose coefficients
are the elements of the vector C. If C has N+1 components,
the polynomial is C(1)*X^N + ... + C(N)*X + C(N+1).
See also POLY, RESIDUE, FZERO.
POLYVALM Evaluate polynomial with matrix argument.
Y = POLYVAL(P,X), when P is a vector of length N+1 whose elements
are the coefficients of a polynomial, is the value of the
polynomial evaluated with matrix argument X. X must be a
square matrix.
Y = P(1)*X^N + P(2)*X^(N-1) + ... + P(N)*X + P(N+1)*I
See also POLYVAL, POLYFIT.
FZERO Scalar nonlinear zero finding.
X = FZERO(FUN,X0) tries to find a zero of the function FUN near X0.
FUN accepts real scalar input X and returns a real scalar function value F
evaluated at X. The value X returned by FZERO is near a point where FUN
changes sign (if FUN is continuous), or NaN if the search fails.
X = FZERO(FUN,X0), where X is a vector of length 2, assumes X0 is an
interval where the sign of FUN(X0(1)) differs from the sign of FUN(X0(2)).
An error occurs if this is not true. Calling FZERO with an interval
guarantees FZERO will return a value near a point where FUN changes
sign.
X = FZERO(FUN,X0), where X0 is a scalar value, uses X0 as a starting guess.
FZERO looks for an interval containing a sign change for FUN and
containing X0. If no such interval is found, NaN is returned.
In this case, the search terminates when the search interval
is expanded until an Inf, NaN, or complex value is found.
X = FZERO(FUN,X0,OPTIONS) minimizes with the default optimization
parameters replaced by values in the structure OPTIONS, an argument
created with the OPTIMSET function. See OPTIMSET for details. Used
options are Display and TolX. Use OPTIONS = [] as a place holder if
no options are set.
X = FZERO(FUN,X0,OPTIONS,P1,P2,...) allows for additional arguments
which are passed to the function, F=feval(FUN,X,P1,P2,...). Pass an empty
matrix for OPTIONS to use the default values.
[X,FVAL]= FZERO(FUN,...) returns the value of the objective function,
described in FUN, at X.
[X,FVAL,EXITFLAG] = FZERO(...) returns a string EXITFLAG that
describes the exit condition of FZERO.
If EXITFLAG is:
> 0 then FZERO found a zero X
< 0 then no interval was found with a sign change, or
NaN or Inf function value was encountered during search
for an interval containing a sign change, or
a complex function value was encountered during search
for an interval containing a sign change.
[X,FVAL,EXITFLAG,OUTPUT] = FZERO(...) returns a structure
OUTPUT with the number of iterations taken in OUTPUT.iterations.
Examples
FUN can be specified using @:
X = fzero(@sin,3)
returns pi.
X = fzero(@sin, 3, optimset('disp','iter'))
returns pi, uses the default tolerance and displays iteration information.
FUN can also be an inline object:
X = fzero(inline('sin(3*x)'),2);
Limitations
X = fzero(inline('abs(x)+1'), 1)
returns NaN since this function does not change sign anywhere on the
real axis (and does not have a zero as well).
X = fzero(@tan,2)
returns X near 1.5708 because the discontinuity of this function near the
point X gives the appearance (numerically) that the function changes sign at
X.
See also ROOTS, FMINBND, @, INLINE.
examples
x = [1.1 2.3 3.9 5.1]
y = [3.887 4.276 4.651 2.117]
a = polyfit(x,y,length(x)-1)
plot(x,y,'r',x,polyval(a,x),'o')
r = roots(a)
x=
1.1000
2.3000
3.9000
5.1000
4.2760
4.6510
2.1170
1.4385
-2.7477
5.4370
y=
3.8870
a=
-0.2015
r=
5.5608
0.7900 + 2.0565i
0.7900 - 2.0565i
inverse and simultaneous equations – determinants and eigenvalues
determinatns
Linear Simultaneous equations
Solving sets of linear simultaneous equations is a very important problem and can be
solved easily in Matlab in a number of convenient ways. A set of simultaneous
equations can be expressed as
Ax=b
where x and b are column vectors of dimension m and A is a square matrix A with
dimensions mm. The goal is find the values for x
Using the \ operator
x=A\b
Using the inverse matrix operation
x = A-1 b
For example, consider the system of simultaneous equations that represent the current
values in a multi-loop circuit
6.1 i1 + 6.8 i2 + 2.4 i3 = 36.5
4.0 i1 + 2.8 i2 - 1.2 i3 = 5.9
3.2 i1 + 0.6 i2 + 3.2 i3 = 12.5
6.1 6.8
2.4
A = [6.1 6.8 2.4 ; 4 2.8 -1.2 ; 3.2 0.6 3.2]  4.0 2.8  1.2
3.2 0.6
36.5
b = [36.5; 5.9; 12.5]  5.9
3.2
0.6791
x=A/b 
12.5
-0.1386 0.2909 0.2130
inv(A)  0.2382 -0.1695 -0.2422
0.0939 -0.2591 0.1449
4.6673
3.7102
0.6791
x = inv(K) * b 
4.6673
3.7102
i1 = -0.6791, i2 = 4.6673, and i3 = 3.7102
Interpolation
The curve fitting methods described in Chapter 6 can be used for interpolation.
Give an example for curve fitting of data from Kaye and Laby eg resistance of a
tungsten filament with temperature
Interpolation with Spline Functions
ref Kharab & Guenther p153
Numerical Differentiation
Often the derivation of a function has to be estimated from a set of tabulated data. We
will consider techniques for estimating the first and second derivatives.
There are three formula commonly used for estimating the first derivative of a
function f(x)
Numerical Integration
In many cases is impossible to evaluate a definite integral analytically
b
I   f ( x) dx
a
The process of numerical quadrature can be used to find an approximate value for I
by replacing f(x) by an interpolating polynomial and then integrating the polynomial
between the limits a to b.
integration
Bibliography
A. Kharab and R.B. Guenther, An Introduction to Numerical Methods – A Matlab
Approach, (Chapman & Hall/CRC, Boca, 2002)
An introduction to theory and applications in numerical methods for researchers and
professionals with a background only in calculus and basic programming. CD-ROM
contains the Matlab code for many algorithms