[Date Index]
[Thread Index]
[Author Index]
RE: Successive Approximation
*To*: mathgroup at smc.vnet.net
*Subject*: [mg37121] RE: [mg37081] Successive Approximation
*From*: "DrBob" <drbob at bigfoot.com>
*Date*: Thu, 10 Oct 2002 03:21:06 -0400 (EDT)
*Reply-to*: <drbob at bigfoot.com>
*Sender*: owner-wri-mathgroup at wolfram.com
The first problem is that ArcSin[x/10] isn't a left-inverse of 10 Sin[x]
in the region of the root. That's why your code converges on another
root. A better inverse for your purpose is the gi below:
f = #(# + 3) &;
g = 10Sin@# &;
gi = Pi - ArcSin[#/10] &;
Using that inverse, there's still a problem, though. Your
"approximation" of the root with
r = gi@f@r
fails because the derivative of gi@f at the root is
gi'[f@rr]f'[rr]
-2.02342
That's greater than one in magnitude, so distance from the root is
magnified rather than diminished. Instead, try
r = fi@g@r
Where
fi[y_] := Evaluate[x /. Last@Solve[f@x == y, x]]
This should work, since
fi'[g@rr]g'[rr]
-0.49421355442685166
Sure enough, it does work:
Values = NestList[fi[g[#1]] & , 2., 12]
{2., 1.8679332339369226, 1.9368280058437026, 1.9040490698559083,
1.9205018812387413,
1.9124384783620436, 1.916439313493727, 1.9144659935649395,
1.9154421880638148,
1.914959973607326, 1.915198347545482, 1.9150805538598723,
1.9151387724997768}
(f[#1] - g[#1] & ) /@ values
{0.9070257317431825, -0.4688124734947827,
0.2242366717647286, -0.11228304957089463,
0.05509675095190758, -0.02732121417963107,
0.0134795615740817, -0.006667318794729482,
0.003293718665990042, -0.0016281317372435211,
0.0008045637255396088, -0.00039764607942949226,
0.0001965172491082967}
Absolute error is cut in half at each iteration, as expected with a
derivative for fi@g near -1/2. The negative sign causes the error to
alternate in sign. We can also look at the log-absolute value of the
error as follows:
(Log[Abs[f[#1] -
g[#1]]] & ) /@ values
{-0.09758445910053692, -0.7575524337892089, -1.4950532145264737,
-2.18673236744676,
-2.8986645309519163, -3.6000918023816326, -4.306580698204814,
-5.010537479670821,
-5.715738058891756, -6.420322094992857, -7.125210383308283,
-7.829948195959684, -8.534760348785936}
Rest[%] - Drop[%, -1]
{-0.659967974688672, -0.7375007807372648, -0.6916791529202861,
-0.7119321635051565,
-0.7014272714297163, -0.7064888958231816, -0.7039567814660064,
-0.7052005792209357,
-0.7045840361011004, -0.7048882883154262, -0.704737812651401,
-0.7048121528262516}
The difference in logarithm of the absolute error approaches
Log@Abs[fi'[g@rr]g'[rr]]
-0.7047875587967565
If you want to use this for teaching, try to use as little code as
possible -- as I have above -- and always try to avoid Do loops in
Mathematica on principle.
DrBob
-----Original Message-----
From: Flurchick, Kenneth M [mailto:FLURCHICKK at MAIL.ECU.EDU]
To: mathgroup at smc.vnet.net
Subject: [mg37121] [mg37081] Successive Approximation
GentleBeings
I have a straightforward implementation of successive approximations
but I cannot seem to froce the code to find the correct solution when I
have
trig or exponentials involved. Can the assembled wisdom point to
straghtforward fixes I know FindRoot works the object is to teach
programming and successive approx, tho.
Thanks
kenf
Below is the code
Clear[f, g, gi, lim, r, rr, fr, \ gir, a, b, c, d, conv];
Plot[{x * ((x + 3)), 10*Sin[x]}, {x, 0.01, 2.4},
PlotStyle -> {{RGBColor[1, 0, 0], Thickness[ .006]},
{RGBColor[0, 0, 1], Thickness[ .006]}}
];
rr = FindRoot[x * ((x + 3)) == 10*Sin[x], {x, 2, 0.01, 2.4}];
f[a_] := a * ((a + 3)) /; a > 0;
g[b_] := 10. * Sin[b] /; b > 0;
gi[c_] := ArcSin[0.1*c] /; c > 0;
Print["Actual root is ", rr];
lim = 10;
r = 2.0;
conv = 10^-4;
For[i = 1, i < lim, i++,
{
fr = f[r];
gir = gi[fr];
d = Abs[N[gir] - r]; i
If[d < conv, Break[]];
r = gir;
Print["The value of x = ", r, " found after ", i, " iterations,",
" with a tolerence ", d, "\n"]
}
]
Print["The value of x = ", r, " found after ", i, " iterations,",
" with a tolerence ", d, "\n"]
"Every man, woman and responsible child has an unalienable individual,
civil, Constitutional and human right to obtain, own, and carry, openly
or
concealed, any weapon -- rifle, shotgun, handgun, machine gun, anything
--
any time, any place, without asking anyone's permission."
L. Neil Smith
Prev by Date:
** Re: Re: Accuracy and Precision**
Next by Date:
** Re: Re: Accuracy and Precision**
Previous by thread:
**Successive Approximation**
Next by thread:
**Function vs. Procedure**
| |