Fractals/Mathematics/Numerical
It will give you a clear visual perception of the gap between mathematics ... (and) your computer calculations. Lynn Wienck^{[1]}
attractive and repelling Edit
In the terminology of numerical analysis, computation near an attractive fixed point is stable while computation near a repelling fixed point is unstable. Generally speaking, stable computation is trustworthy while unstable computation is not. Mark McClure^{[2]}
perturbation and aproximation Edit
 how solution of the problem is altered for nonzero but small perturbation ( usually denoted as an espilon)
 problem can not be solved explicitely ( using symbolic methods), only numericla aproximation can be found
Convergence Edit
" ... methods of acceleration of convergence. Suppose you have a slowly convergent series, and want to know its sum (numerically). Just by summing x_1 + x_2 + ... + x_{1000} + ... x_{1000000} + ... + x_{1000000000} + ... you will get the required accuracy after 100 years. If you fit your x_n to c_2/n^2 + c_3/n^3 + (a few more terms), you will get the same accuracy of the sum in 1 second." Andrey (theoretical high energy physicist) ^{[3]}
Above problem you can find in parabolic Julia sets.
" your sequence was generated as ... the output of a fixed point iteration . ... The Levin transformation^{[4]} ... uses forward differences successively to remove error terms of an alternating series."^{[5]}
Examples:
 GSL library^{[6]}^{[7]}
 "checking for when the series approximation starts to deviate from the result of running it through the Levin utransform available in the gnu scientific library. so far in my tests this is taking me right up to the event horizon of what will end up being the lowest iteration, even in locations that used to cause the other methods to bail on the series approximation way sooner, but it also does not seem to be going too far so as to cause glitchy artifacts. it seems to be just right, across the board, which i find to be rather exciting! " ( quaz0r about Mandelbrot )^{[8]}
 series calculator ^{[9]}
limitation of native double numbers format Edit
 when 2 pixels separated by less than :
 when the size of a pixel drops below approx ^{[10]}
> 1. + 2 ** 53 == 1.
True
> 10. ** 324 == 0.
True
Precision Edit
Precision in numerical compuiting^{[11]}
How to deal with lack of floating point precision:^{[12]}
 implement higher precision arithmetic than your hardware natively supports
 software emulation ( emulating a double with two floats, fixed points numbers , ...)^{[13]}
 use algorithms that are more numerically stable
 use representable numbers
howmanyfloatingpointnumbersareintheinterval [0,1] ?^{[14]}
Libraries :
 Arb by Fredrik Johansson^{[15]}
See how arb library can be used for checking and adjusting precision :
/*
from arb/examples/logistic.c
public domain.
Author: Fredrik Johansson.
*/
goal = digits * 3.3219280948873623 + 3;
for (prec = 64; ; prec *= 2)
{
flint_printf("Trying prec=%wd bits...", prec);
fflush(stdout);
for (i = 0; i < n; i++)
{
// computation
if (arb_rel_accuracy_bits(x) < goal)
{
flint_printf("ran out of accuracy at step %wd\n", i);
break;
}
}
if (i == n)
{
flint_printf("success!\n");
break;
}
}
How many decimal digits are there in n bits ? Edit
One digit of binary number needs one bit : there are 2 binary digits ( 0 and 1) and bits have 2 states. One digit of decimal number needs approximately 3.4 bits.^{[16]} There are 10 decimal numbers, 3 bits have 8 states which is not enough, 4 bits have 16 states which is too much.
type  Total bits  Bits precision  Number of significant decimal digits 

float  32  24  6 
double  64  53  15 
long double  80  64  18^{[17]} 
mpfr_t^{[18]}  142  41  
mpfr_t  1000  294 
See also: decimalprecisionofbinaryfloatingpointnumbers by RIck Regan
Rules Edit
Rounding Edit
 "compute with two more significant figures than your ultimate answer
 Round only after the last step in calculation. Never do further calculations with rounded numbers.
 for multiplying and dividing find the number of significant digits in each factor. The result will have the smaller number of significant digits.
 For powers and roots, the answer should have the same number of significant digits as the original" ^{[19]}
 "When you add (or subtract), you keep as many decimal places as there are in the least accurate number.
 When you multiply (or divide), you keep as many significant digits as there are in the least accurate number."^{[20]}
See
 gcc froundingmath option
Examples of numerical computings Edit
Seed value Edit
standard images ( without zoom) Edit
 Roundoff Error^{[21]}
 Shadowing lemma : that every pseudoorbit (which one can think of as a numerically computed trajectory with rounding errors on every step) stays uniformly close to some true trajectory^{[22]}
External rays Edit
Parameter rays Edit
The Wolf Jung test : The external parameter rays for angles (in turns)
 1/7 (period 3)
 321685687669320/2251799813685247 (period 51)
 321685687669322/2251799813685247 ( period 51 )
Angles differ by about , but the landing points of the corresponding parameter rays are about 0.035 apart.^{[23]}
Dynamic rays Edit
number type Edit
angle precision Edit
For rotational number ( internal angle) 1/34 ray for external angle :
lands on the alfa fixed point :
It is not a ray for angle :
which land on the point :^{[24]}
Difference between external angles of the rays is :
and between landing points of the rays points is :
/* Maxima CAS code */ (%i1) za:0.491486549841951 +0.091874758908285*%i; (%o1) 0.091874758908285 %i + 0.491486549841951 (%i2) zb:0.508513450158049 +0.091874758908285*%i; (%o2) 0.091874758908285 %i  0.508513450158049 (%i3) abs(zazb); (%o3) 1.0
Escaping test Edit
This test was introduced by by John Milnor.^{[25]} See also analysis by Mark Braverman ^{[26]} and roundoff error by Robert P. Munafo^{[27]}
Comment by Mark McClure :^{[28]} " an escape time algorithm would take forever to generate that type of image, since the dynamics are so slow there. If you want resolution of 1/100, it would take roughly 2*10^8 iterates to move the point z0=0.01 to z=2 by iterating f(z)=z+z5 "
Here's an orbit computed with 1000 decimal digits, escaping after about 300000 iterations by gerrit
Cases Edit
Nonparabolic case Edit
Lets take simple hyperbolic case where parameter c is :
Here repelling fixed point z_f is :
Parabolic case Edit
Lets take simple parabolic case where parameter c is :^{[29]}
Here parabolic fixed point z_f is :
Test Edit
Lets take point z of exterior of Julia set but lying near fixed point :
where n is a positive integer
Check how many iterates i needs point z to reach target set ( = escape) :
where ER is Escape Radius
Show relationship between :
 n
 Last Iteration
 type of numbers used for computations ( float, double, long double, extended, arbitrary precision )
Programs Edit
See FractalForum for evaldraw script^{[30]}
Results Edit
float (24)  double (53)  long double (64)  MPFR (80)  MPFR (100)  

hyperbolic  23  52  63  79  99 
parabolic  12  26  32 
The results for standard C types ( float, double, long double) and MPFR precision are the same for the same precision
1  2  3  4  5  24  53  64  80  100  

hyperbolic  1  2  3  4  5  24  53  64  80  100 
parabolic  3  5  10  19  35  16 778 821 
Relation between number of iterations and time of computation in hyperbolic case :
Using MPFR3.0.0p8 with GMP4.3.2 with precision = 128 bits and Escape Radius = 2.000000 n = 1 distance = 5.0000000000e01 LI = 1 log2(LI) = 0; time = 0 seconds n = 2 distance = 2.5000000000e01 LI = 2 log2(LI) = 1; time = 0 seconds n = 3 distance = 1.2500000000e01 LI = 3 log2(LI) = 2; time = 0 seconds n = 4 distance = 6.2500000000e02 LI = 4 log2(LI) = 2; time = 0 seconds n = 5 distance = 3.1250000000e02 LI = 5 log2(LI) = 2; time = 0 seconds n = 6 distance = 1.5625000000e02 LI = 6 log2(LI) = 3; time = 0 seconds n = 7 distance = 7.8125000000e03 LI = 7 log2(LI) = 3; time = 0 seconds n = 8 distance = 3.9062500000e03 LI = 8 log2(LI) = 3; time = 0 seconds n = 9 distance = 1.9531250000e03 LI = 9 log2(LI) = 3; time = 0 seconds n = 10 distance = 9.7656250000e04 LI = 10 log2(LI) = 3; time = 0 seconds n = 11 distance = 4.8828125000e04 LI = 11 log2(LI) = 3; time = 0 seconds n = 12 distance = 2.4414062500e04 LI = 12 log2(LI) = 4; time = 0 seconds n = 13 distance = 1.2207031250e04 LI = 13 log2(LI) = 4; time = 0 seconds n = 14 distance = 6.1035156250e05 LI = 14 log2(LI) = 4; time = 0 seconds n = 15 distance = 3.0517578125e05 LI = 15 log2(LI) = 4; time = 0 seconds n = 16 distance = 1.5258789062e05 LI = 16 log2(LI) = 4; time = 0 seconds n = 17 distance = 7.6293945312e06 LI = 17 log2(LI) = 4; time = 0 seconds n = 18 distance = 3.8146972656e06 LI = 18 log2(LI) = 4; time = 0 seconds n = 19 distance = 1.9073486328e06 LI = 19 log2(LI) = 4; time = 0 seconds n = 20 distance = 9.5367431641e07 LI = 20 log2(LI) = 4; time = 0 seconds n = 21 distance = 4.7683715820e07 LI = 21 log2(LI) = 4; time = 0 seconds n = 22 distance = 2.3841857910e07 LI = 22 log2(LI) = 4; time = 0 seconds n = 23 distance = 1.1920928955e07 LI = 23 log2(LI) = 5; time = 0 seconds n = 24 distance = 5.9604644775e08 LI = 24 log2(LI) = 5; time = 0 seconds n = 25 distance = 2.9802322388e08 LI = 25 log2(LI) = 5; time = 0 seconds n = 26 distance = 1.4901161194e08 LI = 26 log2(LI) = 5; time = 0 seconds n = 27 distance = 7.4505805969e09 LI = 27 log2(LI) = 5; time = 0 seconds n = 28 distance = 3.7252902985e09 LI = 28 log2(LI) = 5; time = 0 seconds n = 29 distance = 1.8626451492e09 LI = 29 log2(LI) = 5; time = 0 seconds n = 30 distance = 9.3132257462e10 LI = 30 log2(LI) = 5; time = 0 seconds n = 31 distance = 4.6566128731e10 LI = 31 log2(LI) = 5; time = 0 seconds n = 32 distance = 2.3283064365e10 LI = 32 log2(LI) = 5; time = 0 seconds n = 33 distance = 1.1641532183e10 LI = 33 log2(LI) = 5; time = 0 seconds n = 34 distance = 5.8207660913e11 LI = 34 log2(LI) = 5; time = 0 seconds n = 35 distance = 2.9103830457e11 LI = 35 log2(LI) = 5; time = 0 seconds n = 36 distance = 1.4551915228e11 LI = 36 log2(LI) = 5; time = 0 seconds n = 37 distance = 7.2759576142e12 LI = 37 log2(LI) = 5; time = 0 seconds n = 38 distance = 3.6379788071e12 LI = 38 log2(LI) = 5; time = 0 seconds n = 39 distance = 1.8189894035e12 LI = 39 log2(LI) = 5; time = 0 seconds n = 40 distance = 9.0949470177e13 LI = 40 log2(LI) = 5; time = 0 seconds n = 41 distance = 4.5474735089e13 LI = 41 log2(LI) = 5; time = 0 seconds n = 42 distance = 2.2737367544e13 LI = 42 log2(LI) = 5; time = 0 seconds n = 43 distance = 1.1368683772e13 LI = 43 log2(LI) = 5; time = 0 seconds n = 44 distance = 5.6843418861e14 LI = 44 log2(LI) = 5; time = 0 seconds n = 45 distance = 2.8421709430e14 LI = 45 log2(LI) = 5; time = 0 seconds n = 46 distance = 1.4210854715e14 LI = 46 log2(LI) = 6; time = 0 seconds n = 47 distance = 7.1054273576e15 LI = 47 log2(LI) = 6; time = 0 seconds n = 48 distance = 3.5527136788e15 LI = 48 log2(LI) = 6; time = 0 seconds n = 49 distance = 1.7763568394e15 LI = 49 log2(LI) = 6; time = 0 seconds n = 50 distance = 8.8817841970e16 LI = 50 log2(LI) = 6; time = 0 seconds n = 51 distance = 4.4408920985e16 LI = 51 log2(LI) = 6; time = 0 seconds n = 52 distance = 2.2204460493e16 LI = 52 log2(LI) = 6; time = 0 seconds n = 53 distance = 1.1102230246e16 LI = 53 log2(LI) = 6; time = 0 seconds n = 54 distance = 5.5511151231e17 LI = 54 log2(LI) = 6; time = 0 seconds n = 55 distance = 2.7755575616e17 LI = 55 log2(LI) = 6; time = 0 seconds n = 56 distance = 1.3877787808e17 LI = 56 log2(LI) = 6; time = 0 seconds n = 57 distance = 6.9388939039e18 LI = 57 log2(LI) = 6; time = 0 seconds n = 58 distance = 3.4694469520e18 LI = 58 log2(LI) = 6; time = 0 seconds n = 59 distance = 1.7347234760e18 LI = 59 log2(LI) = 6; time = 0 seconds n = 60 distance = 8.6736173799e19 LI = 60 log2(LI) = 6; time = 0 seconds n = 61 distance = 4.3368086899e19 LI = 61 log2(LI) = 6; time = 0 seconds n = 62 distance = 2.1684043450e19 LI = 62 log2(LI) = 6; time = 0 seconds n = 63 distance = 1.0842021725e19 LI = 63 log2(LI) = 6; time = 0 seconds n = 64 distance = 5.4210108624e20 LI = 64 log2(LI) = 6; time = 0 seconds n = 65 distance = 2.7105054312e20 LI = 65 log2(LI) = 6; time = 0 seconds n = 66 distance = 1.3552527156e20 LI = 66 log2(LI) = 6; time = 0 seconds n = 67 distance = 6.7762635780e21 LI = 67 log2(LI) = 6; time = 0 seconds n = 68 distance = 3.3881317890e21 LI = 68 log2(LI) = 6; time = 0 seconds n = 69 distance = 1.6940658945e21 LI = 69 log2(LI) = 6; time = 0 seconds n = 70 distance = 8.4703294725e22 LI = 70 log2(LI) = 6; time = 0 seconds n = 71 distance = 4.2351647363e22 LI = 71 log2(LI) = 6; time = 0 seconds n = 72 distance = 2.1175823681e22 LI = 72 log2(LI) = 6; time = 0 seconds n = 73 distance = 1.0587911841e22 LI = 73 log2(LI) = 6; time = 0 seconds n = 74 distance = 5.2939559203e23 LI = 74 log2(LI) = 6; time = 0 seconds n = 75 distance = 2.6469779602e23 LI = 75 log2(LI) = 6; time = 0 seconds n = 76 distance = 1.3234889801e23 LI = 76 log2(LI) = 6; time = 0 seconds n = 77 distance = 6.6174449004e24 LI = 77 log2(LI) = 6; time = 0 seconds n = 78 distance = 3.3087224502e24 LI = 78 log2(LI) = 6; time = 0 seconds n = 79 distance = 1.6543612251e24 LI = 79 log2(LI) = 6; time = 0 seconds n = 80 distance = 8.2718061255e25 LI = 80 log2(LI) = 6; time = 0 seconds n = 81 distance = 4.1359030628e25 LI = 81 log2(LI) = 6; time = 0 seconds n = 82 distance = 2.0679515314e25 LI = 82 log2(LI) = 6; time = 0 seconds n = 83 distance = 1.0339757657e25 LI = 83 log2(LI) = 6; time = 0 seconds n = 84 distance = 5.1698788285e26 LI = 84 log2(LI) = 6; time = 0 seconds n = 85 distance = 2.5849394142e26 LI = 85 log2(LI) = 6; time = 0 seconds n = 86 distance = 1.2924697071e26 LI = 86 log2(LI) = 6; time = 0 seconds n = 87 distance = 6.4623485356e27 LI = 87 log2(LI) = 6; time = 0 seconds n = 88 distance = 3.2311742678e27 LI = 88 log2(LI) = 6; time = 0 seconds n = 89 distance = 1.6155871339e27 LI = 89 log2(LI) = 6; time = 0 seconds n = 90 distance = 8.0779356695e28 LI = 90 log2(LI) = 6; time = 0 seconds n = 91 distance = 4.0389678347e28 LI = 91 log2(LI) = 7; time = 0 seconds n = 92 distance = 2.0194839174e28 LI = 92 log2(LI) = 7; time = 0 seconds n = 93 distance = 1.0097419587e28 LI = 93 log2(LI) = 7; time = 0 seconds n = 94 distance = 5.0487097934e29 LI = 94 log2(LI) = 7; time = 0 seconds n = 95 distance = 2.5243548967e29 LI = 95 log2(LI) = 7; time = 0 seconds n = 96 distance = 1.2621774484e29 LI = 96 log2(LI) = 7; time = 0 seconds n = 97 distance = 6.3108872418e30 LI = 97 log2(LI) = 7; time = 0 seconds n = 98 distance = 3.1554436209e30 LI = 98 log2(LI) = 7; time = 0 seconds n = 99 distance = 1.5777218104e30 LI = 99 log2(LI) = 7; time = 0 seconds n = 100 distance = 7.8886090522e31 LI = 100 log2(LI) = 7; time = 0 seconds n = 101 distance = 3.9443045261e31 LI = 101 log2(LI) = 7; time = 0 seconds n = 102 distance = 1.9721522631e31 LI = 102 log2(LI) = 7; time = 0 seconds n = 103 distance = 9.8607613153e32 LI = 103 log2(LI) = 7; time = 0 seconds n = 104 distance = 4.9303806576e32 LI = 104 log2(LI) = 7; time = 0 seconds n = 105 distance = 2.4651903288e32 LI = 105 log2(LI) = 7; time = 0 seconds n = 106 distance = 1.2325951644e32 LI = 106 log2(LI) = 7; time = 0 seconds n = 107 distance = 6.1629758220e33 LI = 107 log2(LI) = 7; time = 0 seconds n = 108 distance = 3.0814879110e33 LI = 108 log2(LI) = 7; time = 0 seconds n = 109 distance = 1.5407439555e33 LI = 109 log2(LI) = 7; time = 0 seconds n = 110 distance = 7.7037197775e34 LI = 110 log2(LI) = 7; time = 0 seconds n = 111 distance = 3.8518598888e34 LI = 111 log2(LI) = 7; time = 0 seconds n = 112 distance = 1.9259299444e34 LI = 112 log2(LI) = 7; time = 0 seconds n = 113 distance = 9.6296497219e35 LI = 113 log2(LI) = 7; time = 0 seconds n = 114 distance = 4.8148248610e35 LI = 114 log2(LI) = 7; time = 0 seconds n = 115 distance = 2.4074124305e35 LI = 115 log2(LI) = 7; time = 0 seconds n = 116 distance = 1.2037062152e35 LI = 116 log2(LI) = 7; time = 0 seconds n = 117 distance = 6.0185310762e36 LI = 117 log2(LI) = 7; time = 0 seconds n = 118 distance = 3.0092655381e36 LI = 118 log2(LI) = 7; time = 0 seconds n = 119 distance = 1.5046327691e36 LI = 119 log2(LI) = 7; time = 0 seconds n = 120 distance = 7.5231638453e37 LI = 120 log2(LI) = 7; time = 0 seconds n = 121 distance = 3.7615819226e37 LI = 121 log2(LI) = 7; time = 0 seconds n = 122 distance = 1.8807909613e37 LI = 122 log2(LI) = 7; time = 0 seconds n = 123 distance = 9.4039548066e38 LI = 123 log2(LI) = 7; time = 0 seconds n = 124 distance = 4.7019774033e38 LI = 124 log2(LI) = 7; time = 0 seconds n = 125 distance = 2.3509887016e38 LI = 125 log2(LI) = 7; time = 0 seconds n = 126 distance = 1.1754943508e38 LI = 126 log2(LI) = 7; time = 0 seconds n = 127 distance = 5.8774717541e39 LI = 127 log2(LI) = 7; time = 0 seconds
Parabolic case :
Using MPFR3.0.0p8 with GMP4.3.2 with precision = 100 bits and Escape Radius = 2.000000 n = 1 distance = 5.0000000000e01 LI = 3 log2(LI) = 2; time = 0 seconds n = 2 distance = 2.5000000000e01 LI = 5 log2(LI) = 2; time = 0 seconds n = 3 distance = 1.2500000000e01 LI = 10 log2(LI) = 3; time = 0 seconds n = 4 distance = 6.2500000000e02 LI = 19 log2(LI) = 4; time = 0 seconds n = 5 distance = 3.1250000000e02 LI = 35 log2(LI) = 5; time = 0 seconds n = 6 distance = 1.5625000000e02 LI = 68 log2(LI) = 6; time = 0 seconds n = 7 distance = 7.8125000000e03 LI = 133 log2(LI) = 7; time = 0 seconds n = 8 distance = 3.9062500000e03 LI = 261 log2(LI) = 8; time = 0 seconds n = 9 distance = 1.9531250000e03 LI = 518 log2(LI) = 9; time = 0 seconds n = 10 distance = 9.7656250000e04 LI = 1031 log2(LI) = 10; time = 0 seconds n = 11 distance = 4.8828125000e04 LI = 2055 log2(LI) = 11; time = 0 seconds n = 12 distance = 2.4414062500e04 LI = 4104 log2(LI) = 12; time = 0 seconds n = 13 distance = 1.2207031250e04 LI = 8201 log2(LI) = 13; time = 0 seconds n = 14 distance = 6.1035156250e05 LI = 16394 log2(LI) = 14; time = 0 seconds n = 15 distance = 3.0517578125e05 LI = 32778 log2(LI) = 15; time = 0 seconds n = 16 distance = 1.5258789062e05 LI = 65547 log2(LI) = 16; time = 0 seconds n = 17 distance = 7.6293945312e06 LI = 131084 log2(LI) = 17; time = 0 seconds n = 18 distance = 3.8146972656e06 LI = 262156 log2(LI) = 18; time = 0 seconds n = 19 distance = 1.9073486328e06 LI = 524301 log2(LI) = 19; time = 0 seconds n = 20 distance = 9.5367431641e07 LI = 1048590 log2(LI) = 20; time = 1 seconds n = 21 distance = 4.7683715820e07 LI = 2097166 log2(LI) = 21; time = 0 seconds n = 22 distance = 2.3841857910e07 LI = 4194319 log2(LI) = 22; time = 2 seconds n = 23 distance = 1.1920928955e07 LI = 8388624 log2(LI) = 23; time = 2 seconds n = 24 distance = 5.9604644775e08 LI = 16777232 log2(LI) = 24; time = 6 seconds n = 25 distance = 2.9802322388e08 LI = 33554449 log2(LI) = 25; time = 11 seconds n = 26 distance = 1.4901161194e08 LI = 67108882 log2(LI) = 26; time = 21 seconds n = 27 distance = 7.4505805969e09 LI = 134217747 log2(LI) = 27; time = 42 seconds n = 28 distance = 3.7252902985e09 LI = 268435475 log2(LI) = 28; time = 87 seconds n = 29 distance = 1.8626451492e09 LI = 536870932 log2(LI) = 29; time = 175 seconds n = 30 distance = 9.3132257462e10 LI = 1073741845 log2(LI) = 30; time = 351 seconds n = 31 distance = 4.6566128731e10 LI = 2147483669 log2(LI) = 31; time = 698 seconds n = 32 distance = 2.3283064365e10 LI = 4294967318 log2(LI) = 32; time = 1386 seconds n = 33 distance = 1.1641532183e10 LI = 8589934615 log2(LI) = 33; time = 2714 seconds n = 34 distance = 5.8207660913e11 LI = 17179869207 log2(LI) = 34; time = 5595 seconds n = 35 distance = 2.9103830457e11 LI = 34359738392 log2(LI) = 35; time = 11175 seconds n = 36 distance = 1.4551915228e11 LI = 68719476762 log2(LI) = 36; time = 22081 seconds
Analysis Edit
Maximal n in hyperbolic case is almost the same as the precision of the significand ^{[31]}
In parabolic case maximal is
Last Iteration ( escape time = iteration fro which abs(zn) > ER ) is : in hyperbolic case equal to n :
in parabolic case equal to 2^n :
Time of computations is proportional to number of iterations. In hyperbolic case is is short. In parabolic case grows quickly as number of iterations.
Checking one point if escapes in parabolic case :
 for n = 34 take about one hour ( 5 595 seconds )
 for n = 40 take about one day
 for n = 45 take about one month
 for n = 50 take about one year
Q&A Edit
Why programs fails ? Edit
Cancellation of significant digits^{[32]} and loss of significance (LOS).^{[33]}^{[34]}
The program fails because of limited precision of used number types. Addition of big (zp) and small number (distance) gives number which has more decimal digits then can be saved ( floating point type has only 7 decimal digits). Some of the most right digits are cancelled and iteration goes into an infinite loop.
For example : when using floating point in parabolic case lets take
float cx = 0.25;
float Zpx = 0.5;
float Zx ;
float distance;
float Zx2;
float n = 13;
so
distance = pow(2.0,n); // = 1.2207031250e04 = 0,00012207
It is greater then machine epsilon:^{[35]}
distance > FLT_EPSILON // = pow(2, 24) = 5.96e08 = 0,00000006
so this addition still works :
Zx = Zpx + distance; // adding big and small number gives 0,50012207
After multiplication it gives :
Zx2 = Zx*Zx; // = 0,250122
next step is addition. Because floating point format saves only 7 decimal digits it is truncated to :
Zx = Zx2 + cx; // = 0,500122 = Zp + (distance/2)
Here relative error is to big and
d2= 0.0000000149 // distance*distance
is smaller then FLT_EPSILON/2.0 = 0.0000000596;
Solution : increase precision !
#include <stdio.h>
#include <math.h> /* pow() */
#include <float.h> /* FLT_EPSILON */
#include <time.h>
#include <fenv.h> /* fegetround() */
int main()
{
float cx = 0.25;
/* Escape Radius ; it defines target set = { z: abs(z) > ER }
all points z in the target set are escaping to infinity */
float ER = 2.0;
float ER2;
time_t start, end;
float dif;
ER2= ER*ER;
float Zpx = 0.5;
float Zx; /* bad value = 0.5002; good value = 0.5004 */
float Zx2; /* Zx2=Zx*Zx */
float i = 0.0;
float d; /* distance between Zpx=1/2 and zx */
float d2; /* d2=d*d; */
int n = 13;
d = pow (2.0, n);
Zx = Zpx + d;
d2 = d * d;
time (&start);
Zx2 = Zpx * Zpx + 2.0 * d * Zpx + d2;
printf ("Using c with float and Escape Radius = %f \n", ER);
printf ("Round mode is = %d \n", fegetround ());
printf ("i= %3.0f; Zx = %f; Zx2 = %10.8f ; d = %f ; d2 = %.10f\n", i, Zx, Zx2, d, d2);
if (d2 < (FLT_EPSILON / 2.0) )
{
printf("error : relative error to big and d2= %.10f is smaller then FLT_EPSILON/2.0 = %.10f; increase precision ! \a\n",
d2, FLT_EPSILON / 2.0);
return 1;
}
while (Zx2 < ER2) /* ER2=ER*ER */
{
Zx = Zx2 + cx;
d = Zx  Zpx;
d2 = d * d;
Zx2 = 0.25 + d + d2; /* zx2 = zx * zx = (zp + d) * (zp + d) = zp2 +2 * d * zp + d2 = 2.25 + d + d2 */
i += 1.0;
/* printf("i= %3.0f; Zx = %f; Zx2 = %10.8f ; d = %f ; d2 = %f \n", i,Zx, Zx2, d,d2); */
}
time (&end);
dif = difftime (end, start);
printf ("n = %d; distance = %3f; LI = %10.0f log2(LI) = %3.0f time = %2.0lf seconds\n", n, d, i, log2 (i), dif);
return 0;
}
Explanation in polish^{[36]}
What precision do I need for escape test ? Edit
Why MPFR / GMP is slower than standard library ? Edit
Why parabolic dynamics is so weak ? Edit

Distance to fixed point for various types of local discrete dynamics near fixed point . See how parabolic dynamics is complicated

Orbits near fixed point of fat Douady rabbit Julia set. See that to analyze one orbit one have to include period and petal
Distance estimation Edit
Example ^{[37]}
Zoom Edit
 manipulating numbers below in double format shall be avoided.^{[38]}
What precision do I need for zoom ? ^{[39]}^{[40]}^{[41]}^{[42]}
 Pixel density ^{[43]}
 Pixel spacing is a distance between the centers of each twodimensional pixel
/*
precision based on pixel spacing
code by Claude HeilandAllen
http://mathr.co.uk/
*/
static void dorender(struct view *v, struct mandelbrot_image *image) {
mpfr_div_d(v>pixel_spacing, v>radius, G.height / 2.0, GMP_RNDN);
mpfr_t pixel_spacing_log;
mpfr_init2(pixel_spacing_log, 53);
mpfr_log2(pixel_spacing_log, v>pixel_spacing, GMP_RNDN);
int pixel_spacing_bits = mpfr_get_d(pixel_spacing_log, GMP_RNDN);
mpfr_clear(pixel_spacing_log);
int interior = 1;
int float_type = 1;
if (interior) {
if (pixel_spacing_bits > 50 / 2) {
float_type = 2;
}
if (pixel_spacing_bits > 60 / 2) {
float_type = 3;
}
} else {
if (pixel_spacing_bits > 50) {
float_type = 2;
}
if (pixel_spacing_bits > 60) {
float_type = 3;
}
}
const char *float_type_str = 0;
switch (float_type) {
case 0: float_type_str = "float"; break;
case 1: float_type_str = "double"; break;
case 2: float_type_str = "long double"; break;
case 3: float_type_str = "mpfr"; break;
default: float_type_str = "?"; break;
}
One can check it using such program ( automatic math precision ^{[44]}) :
#include <stdio.h>
#include <gmp.h>
#include <mpfr.h>
/*
what precision of floating point numbers do I need
to draw/compute part of complex plane ( 2D rectangle ) ?
http://fraktal.republika.pl/mandel_zoom.html
https://en.wikibooks.org/wiki/Fractals/Computer_graphic_techniques/2D/plane
https://en.wikibooks.org/wiki/Fractals/Mathematics/Numerical
uses the code from
https://gitorious.org/~claude
by Claude HeilandAllen
view = " cre cim radius"
view="0.75 0 1.5"
view="0.275336142511115 6.75982538465039e3 0.666e5"
view="0.16323108442468427133 1.03436384057374316916 1e5"
gcc p.c lmpfr lgmp Wall
*/
int main()
{
//declare variables
int height = 720;
int float_type ;
int pixel_spacing_bits;
mpfr_t radius;
mpfr_init2(radius, 53); //
mpfr_set_d(radius, 1.0e15, GMP_RNDN);
mpfr_t pixel_spacing;
mpfr_init2(pixel_spacing, 53); //
mpfr_t pixel_spacing_log;
mpfr_init2(pixel_spacing_log, 53);
printf ("radius = "); mpfr_out_str (stdout, 10, 0, radius, MPFR_RNDD); putchar ('\n');
// compute
// int mpfr_div_d (mpfr_t rop, mpfr_t op1, double op2, mpfr_rnd_t rnd)
mpfr_div_d(pixel_spacing, radius, height / 2.0, GMP_RNDN);
printf ("pixel_spacing = "); mpfr_out_str (stdout, 10, 0, pixel_spacing, MPFR_RNDD); putchar ('\n');
mpfr_log2(pixel_spacing_log, pixel_spacing, MPFR_RNDN);
printf ("pixel_spacing_log = "); mpfr_out_str (stdout, 10, 0, pixel_spacing_log, MPFR_RNDD); putchar ('\n');
pixel_spacing_bits = mpfr_get_d(pixel_spacing_log, GMP_RNDN);
printf ("pixel_spacing_bits = %d \n", pixel_spacing_bits);
float_type = 0;
if (pixel_spacing_bits > 40) float_type = 1;
if (pixel_spacing_bits > 50) float_type = 2;
if (pixel_spacing_bits > 60) float_type = 3;
switch (float_type) {
case 0: fprintf(stderr, "render using float \n"); break;
case 1: fprintf(stderr, "render using double \n"); break;
case 2: fprintf(stderr, "render using long double \n"); break;
case 3: fprintf(stderr, "render using MPFR  arbitrary precision \n");
}
return 0;
}
Tent map Edit
Numerical error in computing orbit of the tent map with parameter m = 2. Aftere 50 iterations of double numbers all orbits fall into zero.
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
/*
https://math.stackexchange.com/questions/2453939/isthischaracteristicoftentmapusuallyobserved
gcc t.c Wall
a@zelman:~/c/varia/tent$ ./a.out
*/
/*  constants  */
double m = 2.0; /* parameter of tent map */
double a = 1.0; /* upper bound for randum number generator */
int iMax = 100;
/*  functions  */
/*
tent map
*/
double f(double x0, double m){
double x1;
if (x0 < 0.5)
x1 = m*x0;
else x1 = m*(1.0  x0);
return x1;
}
/* random double from 0.0 to a
https://stackoverflow.com/questions/13408990/howtogeneraterandomfloatnumberinc
*/
double GiveRandom(double a){
srand((unsigned int)time(NULL));
return (((double)rand()/(double)(RAND_MAX)) * a);
}
int main(void){
int i = 0;
double x = GiveRandom(a); /* x0 = random */
for (i = 0; i<iMax; i++){
printf("i = %3d \t x = %.16f\n",i, x);
x = f(x,m); /* iteration of the tent map */
}
return 0;
}
Result:
i = 0 x = 0.1720333817284710 i = 1 x = 0.3440667634569419 i = 2 x = 0.6881335269138839 i = 3 x = 0.6237329461722323 i = 4 x = 0.7525341076555354 i = 5 x = 0.4949317846889292 i = 6 x = 0.9898635693778584 i = 7 x = 0.0202728612442833 i = 8 x = 0.0405457224885666 i = 9 x = 0.0810914449771332 i = 10 x = 0.1621828899542663 i = 11 x = 0.3243657799085327 i = 12 x = 0.6487315598170653 i = 13 x = 0.7025368803658694 i = 14 x = 0.5949262392682613 i = 15 x = 0.8101475214634775 i = 16 x = 0.3797049570730451 i = 17 x = 0.7594099141460902 i = 18 x = 0.4811801717078197 i = 19 x = 0.9623603434156394 i = 20 x = 0.0752793131687213 i = 21 x = 0.1505586263374425 i = 22 x = 0.3011172526748851 i = 23 x = 0.6022345053497702 i = 24 x = 0.7955309893004596 i = 25 x = 0.4089380213990808 i = 26 x = 0.8178760427981615 i = 27 x = 0.3642479144036770 i = 28 x = 0.7284958288073540 i = 29 x = 0.5430083423852921 i = 30 x = 0.9139833152294159 i = 31 x = 0.1720333695411682 i = 32 x = 0.3440667390823364 i = 33 x = 0.6881334781646729 i = 34 x = 0.6237330436706543 i = 35 x = 0.7525339126586914 i = 36 x = 0.4949321746826172 i = 37 x = 0.9898643493652344 i = 38 x = 0.0202713012695312 i = 39 x = 0.0405426025390625 i = 40 x = 0.0810852050781250 i = 41 x = 0.1621704101562500 i = 42 x = 0.3243408203125000 i = 43 x = 0.6486816406250000 i = 44 x = 0.7026367187500000 i = 45 x = 0.5947265625000000 i = 46 x = 0.8105468750000000 i = 47 x = 0.3789062500000000 i = 48 x = 0.7578125000000000 i = 49 x = 0.4843750000000000 i = 50 x = 0.9687500000000000 i = 51 x = 0.0625000000000000 i = 52 x = 0.1250000000000000 i = 53 x = 0.2500000000000000 i = 54 x = 0.5000000000000000 i = 55 x = 1.0000000000000000 i = 56 x = 0.0000000000000000 i = 57 x = 0.0000000000000000 i = 58 x = 0.0000000000000000 i = 59 x = 0.0000000000000000 i = 60 x = 0.0000000000000000 i = 61 x = 0.0000000000000000 i = 62 x = 0.0000000000000000 i = 63 x = 0.0000000000000000 i = 64 x = 0.0000000000000000 i = 65 x = 0.0000000000000000 i = 66 x = 0.0000000000000000 i = 67 x = 0.0000000000000000 i = 68 x = 0.0000000000000000 i = 69 x = 0.0000000000000000 i = 70 x = 0.0000000000000000 i = 71 x = 0.0000000000000000 i = 72 x = 0.0000000000000000 i = 73 x = 0.0000000000000000 i = 74 x = 0.0000000000000000 i = 75 x = 0.0000000000000000 i = 76 x = 0.0000000000000000 i = 77 x = 0.0000000000000000 i = 78 x = 0.0000000000000000 i = 79 x = 0.0000000000000000 i = 80 x = 0.0000000000000000 i = 81 x = 0.0000000000000000 i = 82 x = 0.0000000000000000 i = 83 x = 0.0000000000000000 i = 84 x = 0.0000000000000000 i = 85 x = 0.0000000000000000 i = 86 x = 0.0000000000000000 i = 87 x = 0.0000000000000000 i = 88 x = 0.0000000000000000 i = 89 x = 0.0000000000000000 i = 90 x = 0.0000000000000000 i = 91 x = 0.0000000000000000 i = 92 x = 0.0000000000000000 i = 93 x = 0.0000000000000000 i = 94 x = 0.0000000000000000 i = 95 x = 0.0000000000000000 i = 96 x = 0.0000000000000000 i = 97 x = 0.0000000000000000 i = 98 x = 0.0000000000000000 i = 99 x = 0.0000000000000000
See also Edit
 A few notes on computation by Mark McClure
 The Numerical Tours of Data Sciences, by Gabriel Peyré, gather Matlab, Python, Julia and R experiments to explore modern mathematical data sciences
 Numerical Errors in Logistic Map Calculation
 Keisan Online Calculator
 Finite arithmetic and error analysis in python by Gonzalo Galiano Casas and Esperanza García Gonzalo
 math.stackexchange question: isthereanequationforthenumberofiterationsrequiredtoescapethemandelb ?
References Edit
 ↑ mathoverflow question : /roundingerrorsinimagesofjuliasets
 ↑ ChaosAndFractals by Mark McClure. 2.11 A few notes on computation
 ↑ limit of series at gmane.org discussion
 ↑ math.stackexchange questions : levinsutransformation
 ↑ math.stackexchange questions : acceleratingconvergenceofasequence
 ↑ gsl1.0 : Series Acceleration
 ↑ gsl manual : SeriesAcceleration
 ↑ Fractal Forum : seriesacceleration
 ↑ wolframalpha series calculator
 ↑ Fractalshades documentation
 ↑ What you never wanted to know about floating point but will be forced to find out by volkerschatz
 ↑ stackoverflow : Dealing with lack of floating point precision in OpenCL particle system
 ↑ Heavy computing with GLSL – Part 5: Emulated quadruple precision by Henry Thasler
 ↑ howmanyfloatingpointnumbersareintheinterval [0,1 ? by Daniel Lemire]
 ↑ Arb by Fredrik Johansson
 ↑ Computing π with Chudnovsky and GMP by Beej Jorgensen
 ↑ [:w:Extended precision: Extended precision in wikipedia]
 ↑ [:w:GNU MPFRGNU MPFR in wikipedia]
 ↑ Significant Digits and Rounding Copyright © 2003–2014 by Stan Brown
 ↑ mathforum : Rules for Significant Figures and Decimal Places
 ↑ Roundoff Error Robert P. Munafo, 1996 Dec 3.
 ↑ wikipedia : Shadowing_lemma
 ↑ Wolf Jung's test for precision of drawing parameter rays
 ↑ One can find it using program Mandel by Wolf Jung using Ray to point menu position ( or Y key)
 ↑ Dynamics in one complex variable: introductory lectures version of 9591 Appendix G by John W. Milnor
 ↑ Parabolic Julia Sets are Polynomial Time Computable Mark Braverman
 ↑ Roundoff Error by Robert P. Munafo, 1996 Dec 3.
 ↑ math.stackexchange questions : whatistheshapeofparaboliccriticalorbit
 ↑ Parabolic Julia Sets are Polynomial Time Computable by Mark Braverman
 ↑ fractalforums : numericalproblemwithbailouttest
 ↑ wikipedia :Floating point , EEE_754
 ↑ wikipedia : Significant figures
 ↑ wikipedia : Loss of significance
 ↑ IEEE Arithmetic from Numerical Computation Guide by Oracle
 ↑ Machine epsilon
 ↑ Dyskusja po polsku na pl.comp.os.linux.programowanie
 ↑ Precision and mandel zoom using DEM/M
 ↑ Fractalshadesdoc : math
 ↑ reenigne blog : arbitrary precision mandelbrot sets
 ↑ hpdz : Bignum by Michael Condron
 ↑ fractint : Arbitrary Precision and Deep Zooming
 ↑ chaospro documentation : parmparm
 ↑ Pixel density
 ↑ Automatic MathPrecision Robert P. Munafo, 1993 Apr 15.