Computer Applications Lab Computer Applications Lab Lab 9 Lab 9 - - PowerPoint PPT Presentation
Computer Applications Lab Computer Applications Lab Lab 9 Lab 9 Numerical Calculus and Symbolic Numerical Calculus and Symbolic Processing Processing Chapter 8 Chapter 8 - Sections 8.1 through 8.3 Sections 8.1 through 8.3 Chapter 10
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Integration of function f(x) in the interval a ≤ x ≤
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Numerical integration can be approximated by
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Function Description trapz(x,y) Uses trapezoidal integration to compute the integral
function values at the points contained in the array x. quad(‘func’,a,b,tol) Uses an adaptive Simpson’s rule to compute the
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quad(‘func’,a,b,tol) Uses an adaptive Simpson’s rule to compute the integral of the function ’fun’ with a as the lower integration limit and b as the upper limit. The parameter tol is optional. tol indicates the specified error tolerance. quad1(‘fun’,a,b,tol) Uses Lobatto quadrature to compute the integral of the function ’fun’. The rest of the syntax is identical to quad.
The quad and quadl functions are more accurate than
Example: the integration can be computed in
sin(x)dx 2
π
=
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Function Result quad(‘sin’,0,pi) 2 quad1(‘sin’,0,pi) 2 x = 0:0.01:pi trapz(x,sin(x)) 1.9797
The derivative of a function can be interpreted
Three methods can be used to estimate the slope at
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Matlab provides the diff function for computing
Syntax
Example:
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Example: compare the true derivative of sin(x) with backward
and central derivatives.
x = [0:pi/50:2*pi] ; n = length(x); y = sin(x);
% true derivative
td = cos(x) ;
td = cos(x) ;
% numerical derivative backward difference
nd = diff(y) ./ diff(x) ;
% numerical derivative using central difference
ncd = (y(3:n)-y(1:n-2))./ (x(3:n)-x(1:n-2)) ; plot(x,td,'b'), ;hold on plot(x(2:n),nd,'ro'), plot(x(3:n),ncd,'g*') ; xlabel('x'), ylabel('Derivative of SIN(x)') legend('COS(x)','Backward Derivative','Central Derivative')
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0.2 0.4 0.6 0.8 1 SIN(x) COS(x) Backward Derivative Central Derivative
10 1 2 3 4 5 6 7
x Derivative of S
Command Description d = diff(x) Returns a vector d containing the differences between adjacent elements in the vector x. b = polyder(p) Returns a vector b containing the coefficients of the derivative
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b = polyder(p1,p2) Returns a vector b containing the coefficients of the polynomial that is the derivative of the product of the polynomials represented by p1 and p2. [num, den] = polyder(p2,p1) Returns the vectors num and den containing the coefficients of the numerator and denominator polynomials of the derivative
F(x) = x2 + 2x + 1 , G(x) = x3 + 5x2 – x + 8, find the
>> F = [1,2,1], G = [1,5,-1,8];
>> d = polyder(F,G) d = 5 28 30 22 15
>> [num,denum] = polyder(F,G) num = -1 -4 -14 6 17 denum = 1 10 23 6 81 -16 64
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To define a symbol in Matlab use the sym() or the
Examples
When mathematical operations are used with symbols, the result
is symbolic.
Example
>> x = sym(‘sqrt(2)’); >> 2 * x ans = 2*2^(1/2) >> class(ans) ans = sym
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We can use symbols to define symbolic expressions. We can use the
Example
>> syms x y ; >> r = sqrt(x^2+y^2) r = (x^2+y^2)^(1/2)
If we later assign x and y to 3 and 5, typing r will not result in the evaluation
We can use the vector and matrix notation with symbols. Example
>> n = 3 , syms x, A = x.^((0:n)’*(0:n)) A = [ 1, 1, 1, 1] [ 1, x, x^2, x^3] [ 1, x^2, x^4, x^6] [ 1, x^3, x^6, x^9] >> class(A) ans = sym
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The function findsym(E,n) returns the n symbolic variables in
expression E that are closest to x (default symbol in Matlab), with the tie breaker going to the variable closer to z.
>>syms b x1 y >>findsym(6*b+y) ans = b,y >>findsym(6*b+y,1) %Find the one variable closest to >>findsym(6*b+y,1) %Find the one variable closest to x ans = y >>findsym(6*b+y+x1,1) %Find the one variable closest to x ans = x1
>>findsym(6*b+y*i) %i is not symbolic ans = b, y
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With symbolic expressions in Matlab, we can expand, collect, simplify, and
create new expressions from old expressions using the mathematical
The function collect(E) collects coefficients of like powers in the
expression E. If there is more than one variable, you can use the optional form collect(E,v), which collects all the coefficients with the same power of v. power of v.
>>syms x y >>E = (x-5)^2+(y-3)^2; >>collect(E) % will collect x powers ans = x^2-10*x+25+(y-3)^2 >>collect(E,y) % will collect y powers ans = y^2-6*y+(x-5)^2+9
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The expand and simplify functions.
>>syms x y >>expand((x+y)^2) % applies algebra rules ans = x^2+2*x*y+y^2 >>expand(sin(x+y)) % applies trig identities ans = sin(x)*cos(y)+cos(x)*sin(y) >>simplify(6*((sin(x))^2+(cos(x))^2))
% applies another trig identity ans = 6
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The factor function finds the factors of the expression.
>>syms x y >>factor(x^2-1) ans = (x-1)*(x+1)
>>syms x y >>E = x^2+6*x+7; >>F = subs(E,x,y) F = y^2+6*y+7
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The function poly2sym(p) converts the
The function sym2poly(E) converts the
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>> sys x ; >> sym2poly(9*x^2+4) ans = 9 0 4
The function [num,den] = numden(E) returns two
>> syms x >> E1 = x^2 + 5 ; >> E2 = 1 / (x+6) ; >> E2 = 1 / (x+6) ; >> [num,den] = numden(E1+E2) num = x^3 + 6*x^2 + 5*x + 31 den = x + 6 >> pretty(num) % provides mathematical friendly
expression
3 2 x + 6 x + 5 x + 31
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Use the subs and double functions to evaluate an
% evaluate symbolic expressions >>syms x >>E = x^2+6*x+7; >>G = subs(E,x,2) G = 23 >>class(G) ans = double
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% evaluate constant expressions >> sqrt2 = sym(sqrt(2)); >> y = 6 * sqrt2 y = 6 * 2^(1/2) >> double(y) ans = 8.4853
The Matlab function ezplot(E) generates a
The optional form ezplot(E,[xmin
The ezplot function accepts E as string also. The ezplot function does not accept the
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40 50 60 70 80 90 x2-6 x+7
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2 4 6 10 20 30 x
>> eq1 = ‘x+5’ >> solve(eq1) ans = -5 >> solve(‘exp(2*x)+3*exp(x)=54’) ans = [log(-9)] [log(6) ] >> eq3 = ’x^2+9*y^4=0’; >>solve(eq3) %Note that x is presumed to be the unknown variable ans = [ 3*i*y^2] [-3*i*y^2]
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We can solve an equation in terms of a symbol Example:
>> solve (‘b^2+8*c+2*b=0’) ans = -1/8*b^2-1/4*b >> solve (‘b^2+8*c+2*b=0’,’b’) % solve for b ans = [-1+(1-8*c)^(1/2)] [-1-(1-8*c)^(1/2)]
We can solve more than equation
>> eq1 = ‘6*x+2*y=14’; >> eq2 = ‘3*x+7*y=31’; >> solve(eq1,eq2) ans = x: [1x1 sym] % the answer is a structure y: [1x1 sym] >> x = ans.x x = 1 >> y = ans.y y = 4
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Example
>> syms w x y z ; >> M = w - x^2 + log(y) + 5*z ; >> % evaluate M with y = 5 and z = 1 >> N = subs(M,[y,z],[5 1]) ans = w-x^2+log(5)+5 ans = w-x^2+log(5)+5 >> % solve in terms of w i.e. solve for x >> G = solve(N,x) % the result is symbolic array G = (w+log(5)+5)^(1/2)
>> % evaluate the first root if w = 6 >> subs(G(1),w,6) ans = 3.5510
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Example: find the intersection of y = x2 and y = 2x
>> syms x ; >> y1 = x^2 ; >> y2 = 2*x ; >> solve(y1-y2) ans = ans = 2
Example: if x + 6y = a and 2x – 3y = 9, then find the solution for x and y in
terms of the parameter a. >> syms a x y ; >> [x,y] = solve('x+6*y = a','2*x-3*y = 9') x = 1/5*a+18/5 y = 2/15*a-3/5
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We can use the diff function to perform symbolic
>> diff(‘sin(x*y)’) % differentiation with respect to
x
ans = cos(x*y)*y >> syms x y ; % differentiation with respect to y >> diff(sin(x*y),y) ans = x*cos(x*y) >> diff(x*sin(x*y),y,2) % Computes the second derivative ans = -x^3*sin(x*y)
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We can use the int function to perform symbolic
>> int(2*x) % x is defined as a symbol ans = x^2 >> int(sin(x*y),y) % integration with respect to y >> int(sin(x*y),y) % integration with respect to y ans = -1/x*cos(x*y) >> int(sin(x*y),y,0,1) % integrates with respect to y over [0,1] ans = -(-1+cos(x))/x >> int(sin(x*y),0,1) %integrates with respect to x
ans = -(-1+cos(y))/y
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We can use the limit function to find the limits of
>> limit(sin(a*x)/x) % limit as x goes to 0 ans = a >> limit(sin(a*x)/x,3) % limit as x goes to 3 >> limit(sin(a*x)/x,3) % limit as x goes to 3 ans = 1/3*sin(3*a) >> limit(sin(a*x)/x,a,3) % limit as a goes to 3 ans = sin(3*x)/x >> limit((1/x),x,0,’right’) % limit as x goes to 0
from right
ans = Inf
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