Fixed%20point%20iteration

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Lecture 5 Fixed point iteration
Download fixedpoint.m From math.unm.edu/plushnik/
375
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fixedpoint.m - solution of nonlinear equation by
fixed point iterations function x,n, xn
fixedpoint(f, x0, tol, nmax) find the root of
equation xf(x) by fixed point method input
f - inline function x0
- initial guess tol - exit
condition f(x) lt tol nmax - maximum
number of iterations output x
- the approximation for the root n
- number of iterations xn - vector
of apporximations x(iter) compute function
at initial guesses f0 f(x0) n 0
begin iterations while ((abs(f0-x0) gt tol)
(n lt nmax)) x0 f0 f0
f(x0) disp('Error f0-x0',num2str(f0-x0))
if f0 x0 x0 is a root, done
break end n n1 xn(n) x0
end if nnmax disp('warning maximum
iterations reached without conversion') end
xx0 disp('Number of iterations n
',num2str(n)) end
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Enter inline functions
gtgt g1inline('1-x3') gtgt g2inline('(1-x)(1/3)')
gtgt g3inline('(12x3)/(13x2)')
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gtgtfixedpoint(g1,0.5,10(-8),10) Error
f0-x0-0.54492 Error f0-x00.63396 Error
f0-x0-0.85998 Error f0-x00.89482 Error
f0-x0-0.9955 Error f0-x00.99662 Error
f0-x0-1 Error f0-x01 Error f0-x0-1 Error
f0-x01 warning maximum iterations reached
without conversion Number of iterations n 10
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gtgt fixedpoint(g2,0.5,10(-8),100) Error
f0-x0-0.20282 Error f0-x00.15148 Error
f0-x0-0.10605 Error f0-x00.077491 Error
f0-x0-0.054795 Error f0-x00.039626 Error
f0-x0-0.028184 Error f0-x00.020281 Error
f0-x0-0.01447 Error f0-x00.010387 Error
f0-x0-0.0074232 Error f0-x00.0053217 Error
f0-x0-0.0038066 Error f0-x00.0027272 Error
f0-x0-0.0019517 Error f0-x00.0013978 Error
f0-x0-0.0010005 Error f0-x00.00071648 Error
f0-x0-0.00051291 Error f0-x00.00036726 Error
f0-x0-0.00026293 Error f0-x00.00018826 Error
f0-x0-0.00013478 Error f0-x09.6501e-005 Error
f0-x0-6.9091e-005 Error f0-x04.9467e-005 Error
f0-x0-3.5416e-005 Error f0-x02.5357e-005 Error
f0-x0-1.8155e-005 Error f0-x01.2998e-005 Erro
r f0-x0-9.3063e-006 Error f0-x06.663e-006 Erro
r f0-x0-4.7705e-006 Error f0-x03.4155e-006 Err
or f0-x0-2.4454e-006 Error f0-x01.7508e-006 Er
ror f0-x0-1.2535e-006 Error f0-x08.9748e-007 E
rror f0-x0-6.4257e-007 Error
f0-x04.6006e-007 Error f0-x0-3.2938e-007 Error
f0-x02.3583e-007 Error f0-x0-1.6885e-007 Error
f0-x01.2089e-007 Error f0-x0-8.6551e-008 Erro
r f0-x06.1968e-008 Error f0-x0-4.4367e-008 Err
or f0-x03.1765e-008 Error f0-x0-2.2743e-008 Er
ror f0-x01.6283e-008 Error f0-x0-1.1658e-008 E
rror f0-x08.3468e-009 Number of iterations n
52
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gtgt fixedpoint(g3,0.5,10(-8),100) Error
f0-x0-0.031106 Error f0-x0-0.0008513 Error
f0-x0-6.1948e-007 Error f0-x0-3.2774e-013 Numbe
r of iterations n 4 gtgt
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Secant method
Download secant02.m And ftest2.m From
math.unm.edu/plushnik/375
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Secant method to find roots for function
ftest2 x00.1 x12.0starting
points abserr10(-14) stop criterion - desired
absolute error istep0 xn1x0 set initial
value of x to x0 xnx1 main loop to find
root disp('Iterations by Secant Method') while
abs(ftest2(xn))gtabserr istepistep1
fnftest2(xn) fn1ftest2(xn1)
disp('f(x)',num2str(fn),' xn',num2str(xn,15))
display value of function f(x)
xtmpxn-(xn-xn1)fn/(fn-fn1) xn1xn
xnxtmp end fftest2(xn)
disp('f(x)',num2str(fn),' xn',num2str(xn,15))
display value of function f(x) disp('number
of steps for Secant algorithm',num2str(istep))
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test function is defined at fourth line
derivative of function is defined at firth
line function f,fderivativeftest2(x)
fexp(2x)x-3 fderivative2exp(2x)1
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gtgt secant02 Iterations by Secant
Method f(x)53.5982 xn2 f(x)-1.4715
xn0.157697583825433 f(x)-1.2804
xn0.206925256821038 f(x)0.46299
xn0.536842578960542 f(x)-0.094954
xn0.449229649271443 f(x)-0.0057052
xn0.464140200867443 f(x)7.5808e-005
xn0.465093357175321 f(x)-5.9571e-008
xn0.465080858161814 f(x)-6.2172e-013
xn0.465080867975924 f(x)-6.2172e-013
xn0.465080867976027 number of steps for Secant
algorithm9 gtgt
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Inclass3
Modify secant02.m and ftest2.m to find root of
e(-x)-x0 by secant method starting at x0.2 and
x1.5
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Answer to inclass3
gtgt secant02 Iterations by Secant
Method f(x)-1.2769 xn1.5 f(x)-0.088702
xn0.624324608254261 f(x)0.012856
xn0.558951914931113 f(x)-0.00013183
xn0.567227412711665 f(x)-1.9564e-007
xn0.567143415251049 f(x)2.9781e-012
xn0.567143290407884 f(x)2.9781e-012
xn0.567143290409784 number of steps for Secant
algorithm6
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Matlab function fzero
X FZERO(FUN,X0), X0 a scalar Attempts to find
a zero of the function FUN near X0. FUN is a
function handle. The value X returned by FZERO
is near a point where FUN changes sign (if FUN
is continuous), or NaN, if the srootfinding search
fails. X FZERO(FUN,X0), X0 a 2-vector.
Assumes that FUN(X0(1)) and FUN(X0(2)) differ in
sign, insuring a root. X FZERO(FUN,X0,OPTIONS).
Solves the equation with default optimization
parameters replaced by values in the string
OPTIONS, an argument created with the OPTIMSET
function.
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Example of fzero use
gtgt options optimset('disp', 'iter', 'tolx',
1.e-15) gtgt fzero(_at_ftest2,0.1 2,options)
Func-count x f(x)
Procedure 2 0.1 -1.6786
initial 3 0.157698 -1.4715
interpolation 4 0.556708
0.601452 interpolation 5
0.440938 -0.143633 interpolation 6
0.463256 -0.0110609
interpolation 7 0.465084 2.0255e-005
interpolation 8 0.465081
-3.08857e-008 interpolation 9
0.465081 -8.61533e-014 interpolation 10
0.465081 0
interpolation Zero found in the interval 0.1,
2 ans 0.4651 gtgt
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Inclass4
Modify ftest2.m to find root of e(-x)-x0 by
Brents method starting at x0.2 and x1.5
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Answer to inclass4
gtgt fzero(_at_ftest2b,0.2 1.5,options)
Func-count x f(x)
Procedure 2 0.2 0.618731
initial 3 0.624325 -0.0887015
interpolation 4 0.571121
-0.0062285 interpolation 5
0.567143 1.13316e-006 interpolation 6
0.567143 -8.15018e-010
interpolation 7 0.567143 -1.11022e-016
interpolation 8 0.567143
-1.11022e-016 interpolation Zero found
in the interval 0.2, 1.5 ans 0.5671