The graphs of polynomials of order 1 (linear equations) gives a straight ... Gradient. The gradient of a straight line is a measure of its slope with respect to the x-axis. ... quadratic, either by factorisation or by using formulae ..... Set each factor = 0 and solve for x. ... Given a vector r of roots you can create a polynomial ,p, with.
CM2202: Scientific Computing and Multimedia Applications General Maths: 1. Polynomials Prof. David Marshall School of Computer Science & Informatics
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Definitions: General Form, Degree/Order of a Polynomial
The general form of a polynomial function is: f (x) = ax n + bx n−1 + cx n−2 + · · · + k Assuming a 6= 0 the f (x) is said to be a polynomial of degree (or order) n in the variable x.
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Example: Polynomial Degree/Order Examples of polynomials are:
f (x) = 3x 4 − 2x 3 + 5, Polynomial of degree (order) 4 f (x) = x 5 − 23 + x, Polynomial of degree 5 f (x) = x 2 − 4 Polynomial of degree 2 f (x) = 2x − 1 Polynomial of order 1. The order of a polynomial is the highest power of x in the function. 3 / 70
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Roots of a Polynomial: Graphs and Factorisation If we consider the equation f (x) = 0 (f (x) a polynomial of order n) Solving f (x) for x, this will have up to n roots. Polynomials can be represented as graphs or sketches. The roots of a polynomial are the points where the graph crosses the x-axis. In practice, we find these by writing the polynomial as an equation f (x) = 0, and factorising. 4 / 70
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Polynomial Order 1 Graph Example The graphs of polynomials of order 1 (linear equations) gives a straight line. e.g. f (x) = 2x − 1 = 0 180 160 140 120 100 80 60 40 20 0 0
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Polynomial Order 1 Root Example The root of f (x) = 2x − 1
Root
0
0.5
1
Clearly we can trivially solve this analytically, x = 0.5 6 / 70
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Polynomial Order 2 Graph Example The graph of polynomials of order 2 (quadratic equations): e.g. f (x) = x 2 − 4 f(x) = x2 − 4 20
15
10
5
0
−5 −5
−4
−3
−2
−1
0
1
2
3
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5
with roots clearly at x = −2 and x = 2. This shape is also called a parabola 7 / 70
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Polynomial Order 3 Graph Example The graph of polynomials of order 3 gives a cubic curve. e.g. f (x) = x 3 − 2x + 1 f(x) = x3 − 2x +1 20
15
10
5
0
−5
−10
−15
−20 −3
−2
−1
0
1
2
3
(We’ll worry about finding its roots a little later) 8 / 70
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Polynomial Order 4 Graph Example The shape of the quartic curve f (x) = x 4 − 3x 3 + 2x 2 − 2x + 1 looks like this. f(x) = x4 −3x3 + 2x2 −2x +1
180 160 140 120 100 80 60 40 20 0 −3
−2
−1
0
1
2
3
(We’ll worry about finding its roots a little later) 9 / 70
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The Equation of a Straight Line The equation of a straight line is given by: (f (x) =)y = mx + c where m represents the slope of the line (i.e. the gradient ) and c is the point where the line crosses the y -axis (i.e. the y -intercept ). The point where the line crosses the x-axis is called the x-intercept. 180 160 140 120 100 80 60 40 20 0 0
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E.g.: Polynomial Order 1: f (x) = 2x − 1 above. 10 / 70
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Gradient The gradient of a straight line is a measure of its slope with respect to the x-axis. Gradient is defined as rate of inclination: the difference in y , dy divided by the difference in x, dx, between one point and another point on the line. A positive gradient indicates an ’uphill’ slope with respect to the positive direction of the c x-axis. A negative gradient indicates a ’downhill’ slope with respect to the positive direction of the x-axis.
y = mx + c
dy
y θ dx m = tan(θ) =
x
dy dx
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Calculating the Gradient The gradient of a line may be found from any two points on a line. In general the gradient, m of the line passing through A(x1 , y1 ) and B(x2 , y2 ) is : m=
B(x2 , y2 )
y2 − y1 dy = dx x2 − x1
m is also the tangent of the angle the line makes with the x-axis.
m = tan(θ) =
y = mx + c
dy
θ A(x1 , y1 )
dx m = tan(θ) =
dy dx
=
y2 −y1 x2 −x1
dy y2 − y1 = dx x2 − x1
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Quadratic equations A equation of the form ax 2 + bx + c = 0 is called a quadratic e.g. x 2 − 3x + 2 = 0 f(x) = f(x) = x2 − 3x + 2 45 40 35 30 25 20 15 10 5 0 −5 −5
−4
−3
−2
−1
0
1
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5
Notice that the graph crosses the x-axis at two points x = 1 and x = 2. These values are the roots of the quadratic equation x 2 − 3x + 2 = 0 13 / 70
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Finding Roots of a Quadratic Equation The roots of a quadratic equation can also be found by solving the quadratic, either by factorisation or by using formulae Factorisation example: x 2 − 3x + 2 = 0 (x − 1)(x − 2) = 0 This means that either x − 1 = 0 or x − 2 = 0 Therefore the roots of this equation are x = 1 and x = 2
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General quadratic equation solution by formula The roots of the general quadratic polynomial ax 2 + bx + c with a 6= 0 are given by the formula √ −b ± b 2 − 4ac x= 2a Note if a = 0 then the equation would not be a quadratic! So for x 2 − 3x + 2 = 0, a = 1, b = −3 and c = 2, and so b 2 − 4ac = −3 × −3 − 4 × 1 × 2 = 9 − 8 = 1 Therefore x
= =
√ −(−3) ± 1 3±1 = 2 2 4 2 or = 2 or 1 as above. 2 2 15 / 70
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Using the Discriminant to determine the nature of the roots of a quadratic Note: The value of the expression will determine the nature of the roots of the equation, and it is called the discriminant. If the discriminant is zero, i.e. b 2 − 4ac = 0 , the roots of the −b equation are x = −b 2a , 2a — Two equal roots. If the discriminant is positive i.e. b 2 − 4ac > 0 the roots are unequal √and √ 2 2 x = −b+ 2ab −4ac or x = −b− 2ab −4ac — Two distinct real roots. If the discriminant is negative i.e. b 2 − 4ac < 0 there are no real roots of the equation, i.e. the roots are complex (see later) 16 / 70
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The 3 possible cases visualised f(x) = x2 − 4
f(x) = x2
25
f(x) = 3x2 + x + 2 80
20 70
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20
5
0 10
0 −5
−4
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0
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−5 −5
0 −5
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−1
0
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(i) Two equal roots (ii) Two distinct roots (iii) No real roots The above diagrams of ax 2 + bx + c (a 6= 0) show the three possible cases: In (i) the curve touches the x-axis, i.e. for y = 0 there are two equal values of x. In (ii) the curve cuts the x-axis, i.e. for y = 0 there are two real distinct values of x. In (iii) the does not cut the x-axis, i.e. for y = 0 there is no real values of x. 17 / 70
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Exercise: Quadratic equations (1) Without solving the equation 4x 2 − 12x − 9 = 0 , determine the nature of its roots.
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Exercise: Quadratic Equations (2) Solve the following quadratic equation: 4x 2 − 8x − 9 = 0
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Exercise: Quadratic Equations(3) Given that the quadratic equation (k 2 − 1)x 2 − 2kx + 3k + 1 = 0 (in x) has equal roots, show that 3k 3 − 3k − 1 = 0.
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Curve Sketching
In this section we will look at How we sketch or plot polynomials and other functions. Properties of functions that may be observed from their graph. Special case functions — rational functions.
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Curve Sketching: Straight Lines (1) Graph of a straight line y = mx + c 180 160 140 120 100 80 60 40 20 0 0
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c is the point at which the line cuts the y-axis m is the gradient Need only find 2 points to plot a straight line.
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Curve Sketching Example: Straight Lines (2) Sketch y = 2x − 6 From the equation (value of c) we know that the line will cross the y -axis at coordinates (0, −6) To find the second point, let’s find out where the line cuts the x-axis: Let y = 0 then solve 2x − 6
=
0
2x
=
6
x
= 3
Therefore the second point has coordinates (3, 0)
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Curve Sketching Example: Straight Lines (2) So we can plot: y = 2x − 6 5
0 (−c/m=3,0)
−5 (0,c=−6)
−10
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−20
−6
−4
−2
0 x
2
4
6
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Curve Sketching: Quadratic Equations (1) Graph of a quadratic y = ax 2 + bx + c: If a < 0 then we get hill or frown shape curve. If a > 0 then we get valley or smile shape curve. f(x) = −2x3 −1x2 +20
f(x) = 2x3 −1x2 −20
200 180
0
160 140 120
−50
100 80 −100
60 40 20
−150
0 −20 −200 −10
−8
−6
−4
−2
0
2
4
6
8
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−10
−8
−6
−4
−2
0
2
4
6
8
10
Find where the curve cuts the x-axis by factorising (coming very shortly) (let y = 0). The curve cuts the y -axis at c (by setting x = 0) Later we will look at how you can investigate the nature of turning points and find their co-ordinates. (See Calculus: differentiation) 25 / 70
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Odd and Even functions Any function for which f (−x) = f (x) is called an even function. An example of an even function is
f(x) = x 2 : f(−3) = 9 = f (3). Any function for which f (−x) = −f (x) is called an odd function. An example of an odd function is
f(x) = x 3 : f(−3) = −27 = −f (3) In general, functions f (x) = x n have the property that f (−x) = f (x) for even values of n f (−x) = −f (x) for odd values of n. 26 / 70
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Exercise: Odd and Even functions
Show that the function y = f (x) = 4x 3 − x is an odd function.
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Visualising Odd and Even functions From the sketch of a graph we can notice the following: The graph of an even function will be symmetrical about the y -axis. The graph of an odd function will be unchanged under a 180◦ rotation about the origin. Examples: f(x) = x2
f(x) =4*x3 − x 250
100 200 150
80 100 50
60
0
40
−50 −100
20 −150 −200
0 −250 −4
−3
−2
−1
0
1
2
3
Odd Function
4
−10
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0
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Even Function 28 / 70
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Is the following Graph Odd or Even? sin(x)
Odd or Even? 29 / 70
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And is this Graph Odd or Even? cos(x)
Odd or Even? 30 / 70
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Trigonometric Functions: sin(x) sin(x)
sin(x) is odd Clearly: sin(−x) = − sin(x) 31 / 70
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Trigonometric Functions: cos(x) cos(x)
cos(x) is even Clearly: cos(−x) = cos(x)
Note : These results will be important later (see Complex Numbers and Fourier Theory) 32 / 70
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Odd or Even:
Are the following functions odd or even: f (x) = − sin(x)? f (x) = − cos(x)? f (x) = sin(x)2 ? f (x) = cos(x)2 ? f (x) = cos(x) + sin(x)?
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Rational Functions Not all functions are polynomials or even continuous. Rational Functions: A function which can be expressed as D(x) are both polynomials.
N(x) D(x)
where N(x) and
For example: f (x) =
(x + 1) (x − 1)
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Example: Graph of f (x) =
Rational Functions
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(x+1) (x−1) (x + 1)/(x − 1)
4
3
2
1
0
−1
−2
−6
−4
−2
0 x
2
4
6
Notice this graph tends to ∞ at x =. Such a line is called an Asymptote A straight line towards which a curve approaches but never meets 35 / 70
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Curve Sketching Revisited (for Rational Functions) The five basic investigations for curve sketching are 1
Symmetry (or even function) (if obvious) does f (x) = f (−x)?
2
Intersection with axes, does the curve cross the x-axis or y -axis?
3
Behaviour of f (x) as x → ∞ and as x → ±0
4
Where is f (x) undefined? Where are the poles? where the denominator is zero and the function shoots off to infinity (∞)
5
Where are the maximum and minimum points (this will be covered later when we look at Calculus: differentiation)
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Example: Rational Function Curve Sketching Sketch: f (x) =
2x + 1 (x − 2)(x + 1)(x − 3)
To sketch a graph like this we Factorise the numerator and denominator.
We have pre-factorised f (x) in this example. A little more on factorisation to follow soon. And then use the basic investigations: Is the function even (is the graph symmetrical)? Where does f (x) intersect with the axes What happens to the function when x becomes very large ±, i.e. x → ±∞ ? What happens to the function when x gets close to zero (from ±)? Where are the poles — the points where the denominator is zero? 37 / 70
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Example: f (x) =
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2x+1 (x−2)(x+1)(x−3)
In this example we can see that: It has poles (goes to infinity) when x = −1, x = 2 and x = 3: The places where the denominator equals zero Easily obtained from the factors of the denominator Set each factor = 0 and solve for x.
These are examples of vertical asymptotes. (2 x + 1)/((x + 1) (x − 2) (x − 3)) 2.5
2
1.5
1
0.5
0
−0.5
−1
−4
−2
0
2
4
6
x
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Exercise: Rational Functions Consider the function: f (x) =
x (1 − x)
Where is the pole? Where does it intersect with the x and y axes? What happens for ± large values of x, x → ±∞? Roughly sketch the graph of f (x).
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Polynomials/Equations and Mathematical Functions in MATLAB MATLAB provides some excellent and easy to use tools to help us to Represent and manipulate polynomials More generally, represent and manipulate almost any mathematical expressions or equations Easy way to plot graphs of mathematical equations/expressions Easy way to factorise mathematical expressions Easy way to solve equations More advanced maths - see later 40 / 70
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Polynomials in MATLAB MATLAB provides some basic functions for declaring and manipulating polynomials. This is part of basic MATLAB — not an additional toolbox Some (not all) MATLAB functions: poly(r), Create/return a polynomial whose roots are r. r is a vector whose elements are the coefficients of the polynomial whose roots are the elements of r .
roots(p), find the roots of a polynomial, p. polyval(p,x) returns the value of a polynomial p evaluated at x. x can be a single value or a MATLAB vector of several values
For full information see MATLAB documentation: Functions → Mathematics → Polynomials and help poly etc. 41 / 70
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MATLAB Polynomial Representation and Creation
In MATLAB: A polynomial variable p is a vector of length n + 1 whose elements are the coefficients in descending powers of the polynomial to be evaluated: f (x) = p1 x n + p2 x n−1 + . . . + pn x + pn+1
There are two ways to create p : 1
via its roots
2
via its coefficients
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MATLAB Polynomial Creation: poly() Given a vector r of roots you can create a polynomial ,p, with poly(r): poly create eg.m >> r = [-1,1]; % Define roots >> p = poly(r) % Create a polynomial with roots p = 1 0 -1 >> r1 = roots(p) % Solve the polynomial r1 = -1 1
Here we have a created a polynomial f (x) = x 2 − 1 (the x coefficient element is zero in the vector, p). The function roots(p) returns the roots of a given polynomial, p and solved it. Clearly, poly() and roots() perform opposite roles. 43 / 70
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MATLAB Creating Polynomials: Declaring a poly and evaluating it, polyval() Now we have seen the MATLAB polynomial structure we can simply declare it as a MATLAB vector; p = [1, 0, -1]; % a Polynomial f (x) = x 2 − 1 Evaluating a polynomial,polyval() Given a polynomial p and a point or vector of points, x, the function, polyval(p, x) evaluates them: >> p = [1, 0, -1]; % A Polynomial f (x) = x 2 − 1 >> x = 1; % A data point x >> y = polyval(p,x) % Evaluate p at data point x y = 0 44 / 70
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MATLAB Creating Polynomials: polyval() multiple values
In MATLAB it is easy to create a range of values, x and use these with polyval(): >> p = [1, 0, -1]; % a Polynomial f (x) = x 2 − 1 >> x = 1:3 % A range of points x = 1 2 3 >> y = polyval(p,x) % Evaluate p at data points x y = 0 3 8
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MATLAB: Plotting Polynomial Graphs (1)
The MATLAB plot() function1 plot creates linear x-y plots; If x and y are vectors of the same length, the command: plot(x,y) Opens a MATLAB figure (graphics window) Draws an x-y plot of the elements of x versus the elements of y .
Stay tuned for a lot more details on plotting in MATLAB in Dr. Lai’s Lectures 1
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MATLAB: Plotting Polynomial Graphs (2) Given a set of values x we can evaluate a given polynomial, p using polyval(): poly1.m p = [2, -1]; % Create a polynomial f (x) = 2x − 1 x= -5:100; % Create some x values to evaluate f(x) y = polyval(p,x); % evaluate polynomial plot(x,y); % plot x against y axis tight ; % force concise display of plot roots p = roots(p) % Solve the Equation
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MATLAB: Plotting Polynomial Graphs (3) Running poly1.m we get the following output and figure:
180
poly1.m Output
160 140
>> poly1
120 100 80
roots p =
60 40
0.5000
20 0 0
10
20
30
40
50
60
70
80
90
100
which is the solution we expect from f (x) = 2x − 1 = 0.
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More Polynomial Examples: poly2.m poly2.m f(x) = x2 − 4
% Create a polynomial % f ( x ) = x ˆ2 −4 p1 = [ 1 , 0 , −4]; % C r e a t e some v a l u e s o f x xmin = −5.0; xmax = 5 . 0 ; xstep = 0.2; % Make x d a t a p t s x=xmin : x s t e p : xmax ; % Evaluate polynomial y = p o l y v a l ( p1 , x ) ;
20
15
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5
0
−5 −5
−4
−3
−2
−1
0
1
2
3
4
5
Note: poly2.m contains some more code to make plot prettier.
plot (x , y ); % plot i t 49 / 70
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More Polynomial Examples: poly2 1.m, poly2 2.m, poly2 3.m Some similar examples (different polynomials - Quadratics): poly2 1.m — f (x) = x 2 − 3x + 2
f(x) = f(x) = x2 − 3x + 2 45 40
p = [ 1 , −3, 2 ] ;
35 30
poly2 2.m — f (x) = 3x 2 + x + 2
25 20 15
p = [3 , 1 , 2];
10 5
poly2 3.m — f (x) = x
2
0 −5 −5
p = [1 , 0 , 0];
−4
−3
−2
−1
0
1
2
3
4
5
Output of poly2 1.m
Note: We have seen examples of all these plots in previous slides. 50 / 70
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More Polynomial Examples: poly3.m, poly4.m Some further examples (different polynomials - Cubic and Quartic): f(x) = x3 − 2x +1 20
poly3.m — f (x) = x 3 − 2x + 1
15
10
p = [ 1 , 0 , −2, 1 ] ;
5
0
poly4.m — f (x) = x 4 − 3x 3 + 2x 2 − 2x + 1 p = [ 1 , −3, 2 , −2, 1 ] ;
−5
−10
−15
−20 −3
−2
−1
0
1
2
3
Output of poly3.m
Note: We have seen examples of all these plots in previous slides. 51 / 70
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A Convenient Polynomial Plotting Function: poly plot() poly plot() — poly plot.m [ r o o t s p ] = p o l y p l o t ( p , xmin , xmax , x s t e p , y o f f s e t )
Let’s combine all the handy plotting facilities seen in last examples into a convenient function: Download this function into your MATLAB workspace and type help poly plot or view in text editor. The complete inner workings of this function are not so important for now.
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A Convenient Polynomial Plotting Function: poly plot() Usage poly plot() Usage: type help poly plot >> h e l p p o l y p l o t p o l y p l o t ( p , xmin , xmax , x s t e p , y o f f s e t ) Function to take i n a a r b i t r a r y polynomial , p and p l o t i t o v e r a g i v e n r a n g e and p l o t a l l i t s REAL r o o t s . INPUTS : p − v e c t o r o f p o l y n o m i a l c o e f f i c i e n t s . See man p o l y f o r d e t a i l s xmin − minimum v a l u e o f x r a n g e xmax − maximum v a l u e o f x r a n g e xstep − increment step s i z e for x y o f f s e t − o f f s e t from y v a l u e s f o r plot prettiness . Output : r o o t s p − r o o t s o f t h e p o l y n o m i a l 53 / 70
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poly plot eg.m:
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Calling poly plot() example
poly plot eg.m % Create a polynomial % f ( x ) = 4 x ˆ3 − x p = [4 ,0 , −1 ,0] % C r e a t e some x v a l u e s xmin = −4.0; xmax = 4 . 0 ; xstep = 0.1; % E x t e n d a x e s beyond r a n g e o f y f o r p l o t yoffset = 10.0; % C r e a t e some x v a l u e s x=xmin : x s t e p : xmax ; % S o l v e and P l o t p o l y n o m i a l p r o o t s p = p o l y p l o t ( p , xmin , xmax , x s t e p , y o f f s e t ) 54 / 70
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poly plot eg.m output f(x) =4*x3 − x 250
poly plot eg.m Output
200 150
>> p o l y p l o t e g
100 50 0
ans =
−50 −100
0 0.5000 −0.5000
−150 −200 −250 −4
−3
−2
−1
0
1
2
3
4
Output of poly plot eg.m
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MATLAB Polynomials/Rational Functions: Symbolic Computation MATLAB allows you to do direct algebra via symbolic computation Create algebraic variables and functions Manipulate these algebraically or symbolically Easily solve equations Easily plot graphs A lot more complex operations possible — but not studied here. Other commercial packages also allow similar facilities: e.g. Mathematica and Maple 56 / 70
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MATLAB Symbolic Math Toolbox
MATLAB provides this amazing computation potential via its Symbolic Math Toolbox Additional (add-on) Toolbox Not part of standard MATLAB
Check that your MATLAB licence supports this. Available on School MATLAB Licence.
For full details see MATLAB documentation: Symbolic Math Toolbox and doc sym, help sym or help syms etc.
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Creating a Symbolic Variable You have to create a special symbolic variable in MATLAB. The easiest way is via the syms command: syms x ;
Having created a symbolic variable you can create symbolic expressions, e.g: f = x ˆ2 −2∗x +1;
See doc syms, or help syms etc. for more details. (Also help sym) 58 / 70
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Plotting a Symbolic Expression This could not be simpler via ezplot(): f(x) =− 2*x2 − x + 20
30
20
10
ezplot ( f )
0
−10
−20
−4
−3
−2
−1
0
1
2
3
4
You can use ezplot() with all other MATLAB Graphics functions Such as plot(), axis control and labelling. 59 / 70
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line plot symbolic eg.m
line plot symbolic eg.m % S i m p l e Example S c r i p t t o p l o t a s t r a i g h t syms x ; m=2;
l i n e u s i n g Symbolic Toolbox
c= −6;
y = m∗x +c ; ezplot (y );
% P l o t X and Y a x e s and l a b e l t i t l e ( ’ y = 2x − 6 ’) h o l d on p l o t ([ −6 6 ] , [ 0 0] , ’ − k ’ ) ; % X A x i s p l o t ( [ 0 0] ,[ −20 6] , ’ − k ’ ) ; % Y A x i s axis tight
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line plot symbolic eg.m cont.
line plot symbolic eg.m % Plot c y axis i n t e r c e p t p l o t (0 , c , ’ o ’ , ’ LineWidth ’ , 2 , . . . ’ MarkerEdgeColor ’ , ’ k ’ , . . . ’ MarkerFaceColor ’ , ’ r ’ , . . . ’ MarkerSize ’ , 1 0 ) ; % P l o t −c /m x a x i s i n t e r c e p t p l o t (−c /m, 0 , ’ o ’ , ’ L i n e W i d t h ’ , 2 , . . . ’ MarkerEdgeColor ’ , ’ k ’ , . . . ’ MarkerFaceColor ’ , ’ b ’ , . . . ’ MarkerSize ’ , 1 0 ) ; % label intercepts offset = 0.5; t e x t (0+ o f f s e t , c , ’ ( 0 , c = −6) ’); t e x t (−c /m,0−2∗ o f f s e t , ’(− c /m= 3 , 0 ) ’ ) ;
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line plot symbolic eg.m Output y = 2x − 6 5
0 (−c/m=3,0)
−5 (0,c=−6)
−10
−15
−20
−6
−4
−2
0 x
2
4
6
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Evaluating a Symbolic Expression: subs() Use the subs() command to evaluate a function: syms x ; >> f = x + 1 ; >> s u b s ( f , 1 ) ans =
2
>> s u b s ( f , 1 : 3 ) ans =
2
3
4
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Substitution via subs() subs() can do much more, it can substitute variable and expressions: >> >> >> >>
syms x ; syms y ; f = x + 1; subs ( f , x , y )
ans =
y + 1
>> g = y + 1 ; >> s u b s ( f , x , g ) ans =
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Let’s do some algebra
You can manipulate expressions via subs() but you can also do things directly: >> g = x ; >> g2 = 2∗ x + 1 ; >> g3 = g + g2 ans = 3∗ x + 1
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Simplifying, Factorising, Expanding Expressions There are MATLAB functions to achieve these: >> syms x ; >> f = x ˆ2 −2∗x +1; >> f 1 = s i m p l i f y ( f ) ans = ( x − 1)ˆ2 >> f 2 = f a c t o r ( f ) f2 = ( x − 1)ˆ2 >> expand ( f 2 ) a n s = x ˆ2 − 2∗ x + 1
simplify() does more than just factorise: >> s i m p l i f y ( s i n ( x ) ˆ 2 + c o s ( x ) ˆ 2 ) ans = 1 66 / 70
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Solving Equations: solve()
Simple using solve(): >> syms x ; >> f = x ˆ2 −2∗x +1; >> r o o t s f = s o l v e ( f ) roots f = 1 1
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Symbolic Math Rational Expressions You can define rational and trigonometric (as well as other) expressions: (2 x + 1)/((x + 1) (x − 2) (x − 3))
symbolic eg.m
2.5
2
f 1 = ( x +1)/( x −1);
1.5
1
% Another f 2 = ( 2 ∗ x +1)/(( x −2)∗( x +1)∗( x − 3 ) ) ; ezplot ( f2 )
0.5
0
−0.5
−1
% Trigonometric f3 = s i n ( x )ˆ2 ;
−4
−2
0
2
4
6
x
ezplot(f2) See symbolic eg.m for examples of all Symbolic methods discussed here. 68 / 70
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Converting from Polynomial to Symbolic Representations MATLAB provides convenience functions to convert between both types of polynomial representations. poly2sym(): >> syms x ; >> p = [ 1 , −2, 1 ] ; % C o n v e r t from P o l y p t o s y m b o l i c >> f x = p o l y 2 s y m ( p ) fx =
x ˆ2 − 2∗ x + 1
% S p e c i f y to Symbolic V a r i a b l e >> f t = p o l y 2 s y m ( p , ’ t ’ ) f t = t ˆ2 − 2∗ t + 1
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Converting from Symbolic to Polynomial Representations As expected, sym2poly(), converts from Symbolic to Polynomial: Creates a vector with the coefficients of given symbolic expression. >> syms x ; >> f = 3∗ x ˆ3 + 2∗ x ˆ2 + x ; >> p = s y m 2 p o l y ( f ) p = 3
2
1
0
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