        It will give you a clear visual perception of the gap between mathematics ... (and) your computer calculations. Lynn Wienck^[1]

attractive and repelling

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

how solution of the problem is altered for non-zero but small perturbation ( usually denoted as an espilon)
problem can not be solved explicitely ( using symbolic methods), only numericla aproximation can be found

Convergence

" ... 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 $x_{i+1}=f(x_{i})$ . ... 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 u-transform 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

when 2 pixels separated by less than : $2^{-53}$
when the size of a pixel drops below approx $10^{-308}$ ^[10]

  > 1. + 2 ** -53 == 1.
    True

 > 10. ** -324 == 0.
    True

Precision

Precision in numerical compuiting^[11]

Floating-point arithmetic Accuracy_problems in wikipedia

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

how-many-floating-point-numbers-are-in-the-interval [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 ?

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

Rules

Rounding

"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 -frounding-math option

Examples of numerical computings

Finding roots of equation

Seed value

fractalforums.org: julia-sets-true-shape-and-escape-time

standard images ( without zoom)

Roundoff Error^[21]
Shadowing lemma : that every pseudo-orbit (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

Parameter rays

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 $10^{-15}$ , but the landing points of the corresponding parameter rays are about 0.035 apart.^[23]

Dynamic rays

number type

angle precision

For rotational number ( internal angle) 1/34 ray for external angle :

$t_{a}={\frac {2^{33}}{2^{34}-1}}={\frac {8589934592}{17179869183}}=0.5000000000291038$

lands on the alfa fixed point :

$z_{a}=0.491486549841951+0.091874758908285i$

It is not a ray for angle :

$t_{b}={\frac {1}{2}}=0.5$

which land on the point :^[24]

$z_{b}=-0.508513450158049+0.091874758908285i$

Difference between external angles of the rays is :

$dt=t_{b}-t_{a}=-0.0000000000291038$

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(za-zb);
(%o3)                                 1.0

Storing the angle as an exact rational (e.g. mpq_t using libgmp), allows it to be accurate for more than 53 iterations (the number of bits of accuracy of double precision floating point).  Convert to double only when needing to do sin and cos for finding the target point.  Double is plenty accurate for target point. ( 	Claude Heiland-Allen	)

Escaping test

This test was introduced by by John Milnor.^[25] See also analysis by Mark Braverman ^[26] and roundoff error by Robert P. Munafo^[27]

Julia set z+z^5. Image and src code

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

Nonparabolic case

Lets take simple hyperbolic case where parameter c is :

 $c=0$

Here repelling fixed point z_f is :

 $z_{f}=1$

Parabolic case

Lets take simple parabolic case where parameter c is :^[29]

 $c={\frac {1}{4}}$

Here parabolic fixed point z_f is :

 $z_{f}={\frac {1}{2}}$

Test

Lets take point z of exterior of Julia set but lying near fixed point :

 $distance=2^{-n}$ 

 $z=z_{f}+distance$

where n is a positive integer

Check how many iterates i needs point z to reach target set ( = escape) :

 $|z_{i}|>ER$

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

See FractalForum for evaldraw script^[30]

Results

Maximal n for which program does not fall into fixed point (columns = number types (precision of significand); rows = case)
	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

Relation between : Last iteration, n ( in columns) and case ( in rows)
	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 MPFR-3.0.0-p8 with GMP-4.3.2 with precision = 128 bits and Escape Radius = 2.000000 
n =   1 distance = 5.0000000000e-01 LI =          1 log2(LI) =   0; time =  0 seconds 
n =   2 distance = 2.5000000000e-01 LI =          2 log2(LI) =   1; time =  0 seconds 
n =   3 distance = 1.2500000000e-01 LI =          3 log2(LI) =   2; time =  0 seconds 
n =   4 distance = 6.2500000000e-02 LI =          4 log2(LI) =   2; time =  0 seconds 
n =   5 distance = 3.1250000000e-02 LI =          5 log2(LI) =   2; time =  0 seconds 
n =   6 distance = 1.5625000000e-02 LI =          6 log2(LI) =   3; time =  0 seconds 
n =   7 distance = 7.8125000000e-03 LI =          7 log2(LI) =   3; time =  0 seconds 
n =   8 distance = 3.9062500000e-03 LI =          8 log2(LI) =   3; time =  0 seconds 
n =   9 distance = 1.9531250000e-03 LI =          9 log2(LI) =   3; time =  0 seconds 
n =  10 distance = 9.7656250000e-04 LI =         10 log2(LI) =   3; time =  0 seconds 
n =  11 distance = 4.8828125000e-04 LI =         11 log2(LI) =   3; time =  0 seconds 
n =  12 distance = 2.4414062500e-04 LI =         12 log2(LI) =   4; time =  0 seconds 
n =  13 distance = 1.2207031250e-04 LI =         13 log2(LI) =   4; time =  0 seconds 
n =  14 distance = 6.1035156250e-05 LI =         14 log2(LI) =   4; time =  0 seconds 
n =  15 distance = 3.0517578125e-05 LI =         15 log2(LI) =   4; time =  0 seconds 
n =  16 distance = 1.5258789062e-05 LI =         16 log2(LI) =   4; time =  0 seconds 
n =  17 distance = 7.6293945312e-06 LI =         17 log2(LI) =   4; time =  0 seconds 
n =  18 distance = 3.8146972656e-06 LI =         18 log2(LI) =   4; time =  0 seconds 
n =  19 distance = 1.9073486328e-06 LI =         19 log2(LI) =   4; time =  0 seconds 
n =  20 distance = 9.5367431641e-07 LI =         20 log2(LI) =   4; time =  0 seconds 
n =  21 distance = 4.7683715820e-07 LI =         21 log2(LI) =   4; time =  0 seconds 
n =  22 distance = 2.3841857910e-07 LI =         22 log2(LI) =   4; time =  0 seconds 
n =  23 distance = 1.1920928955e-07 LI =         23 log2(LI) =   5; time =  0 seconds 
n =  24 distance = 5.9604644775e-08 LI =         24 log2(LI) =   5; time =  0 seconds 
n =  25 distance = 2.9802322388e-08 LI =         25 log2(LI) =   5; time =  0 seconds 
n =  26 distance = 1.4901161194e-08 LI =         26 log2(LI) =   5; time =  0 seconds 
n =  27 distance = 7.4505805969e-09 LI =         27 log2(LI) =   5; time =  0 seconds 
n =  28 distance = 3.7252902985e-09 LI =         28 log2(LI) =   5; time =  0 seconds 
n =  29 distance = 1.8626451492e-09 LI =         29 log2(LI) =   5; time =  0 seconds 
n =  30 distance = 9.3132257462e-10 LI =         30 log2(LI) =   5; time =  0 seconds 
n =  31 distance = 4.6566128731e-10 LI =         31 log2(LI) =   5; time =  0 seconds 
n =  32 distance = 2.3283064365e-10 LI =         32 log2(LI) =   5; time =  0 seconds 
n =  33 distance = 1.1641532183e-10 LI =         33 log2(LI) =   5; time =  0 seconds 
n =  34 distance = 5.8207660913e-11 LI =         34 log2(LI) =   5; time =  0 seconds 
n =  35 distance = 2.9103830457e-11 LI =         35 log2(LI) =   5; time =  0 seconds 
n =  36 distance = 1.4551915228e-11 LI =         36 log2(LI) =   5; time =  0 seconds 
n =  37 distance = 7.2759576142e-12 LI =         37 log2(LI) =   5; time =  0 seconds 
n =  38 distance = 3.6379788071e-12 LI =         38 log2(LI) =   5; time =  0 seconds 
n =  39 distance = 1.8189894035e-12 LI =         39 log2(LI) =   5; time =  0 seconds 
n =  40 distance = 9.0949470177e-13 LI =         40 log2(LI) =   5; time =  0 seconds 
n =  41 distance = 4.5474735089e-13 LI =         41 log2(LI) =   5; time =  0 seconds 
n =  42 distance = 2.2737367544e-13 LI =         42 log2(LI) =   5; time =  0 seconds 
n =  43 distance = 1.1368683772e-13 LI =         43 log2(LI) =   5; time =  0 seconds 
n =  44 distance = 5.6843418861e-14 LI =         44 log2(LI) =   5; time =  0 seconds 
n =  45 distance = 2.8421709430e-14 LI =         45 log2(LI) =   5; time =  0 seconds 
n =  46 distance = 1.4210854715e-14 LI =         46 log2(LI) =   6; time =  0 seconds 
n =  47 distance = 7.1054273576e-15 LI =         47 log2(LI) =   6; time =  0 seconds 
n =  48 distance = 3.5527136788e-15 LI =         48 log2(LI) =   6; time =  0 seconds 
n =  49 distance = 1.7763568394e-15 LI =         49 log2(LI) =   6; time =  0 seconds 
n =  50 distance = 8.8817841970e-16 LI =         50 log2(LI) =   6; time =  0 seconds 
n =  51 distance = 4.4408920985e-16 LI =         51 log2(LI) =   6; time =  0 seconds 
n =  52 distance = 2.2204460493e-16 LI =         52 log2(LI) =   6; time =  0 seconds 
n =  53 distance = 1.1102230246e-16 LI =         53 log2(LI) =   6; time =  0 seconds 
n =  54 distance = 5.5511151231e-17 LI =         54 log2(LI) =   6; time =  0 seconds 
n =  55 distance = 2.7755575616e-17 LI =         55 log2(LI) =   6; time =  0 seconds 
n =  56 distance = 1.3877787808e-17 LI =         56 log2(LI) =   6; time =  0 seconds 
n =  57 distance = 6.9388939039e-18 LI =         57 log2(LI) =   6; time =  0 seconds 
n =  58 distance = 3.4694469520e-18 LI =         58 log2(LI) =   6; time =  0 seconds 
n =  59 distance = 1.7347234760e-18 LI =         59 log2(LI) =   6; time =  0 seconds 
n =  60 distance = 8.6736173799e-19 LI =         60 log2(LI) =   6; time =  0 seconds 
n =  61 distance = 4.3368086899e-19 LI =         61 log2(LI) =   6; time =  0 seconds 
n =  62 distance = 2.1684043450e-19 LI =         62 log2(LI) =   6; time =  0 seconds 
n =  63 distance = 1.0842021725e-19 LI =         63 log2(LI) =   6; time =  0 seconds 
n =  64 distance = 5.4210108624e-20 LI =         64 log2(LI) =   6; time =  0 seconds 
n =  65 distance = 2.7105054312e-20 LI =         65 log2(LI) =   6; time =  0 seconds 
n =  66 distance = 1.3552527156e-20 LI =         66 log2(LI) =   6; time =  0 seconds 
n =  67 distance = 6.7762635780e-21 LI =         67 log2(LI) =   6; time =  0 seconds 
n =  68 distance = 3.3881317890e-21 LI =         68 log2(LI) =   6; time =  0 seconds 
n =  69 distance = 1.6940658945e-21 LI =         69 log2(LI) =   6; time =  0 seconds 
n =  70 distance = 8.4703294725e-22 LI =         70 log2(LI) =   6; time =  0 seconds 
n =  71 distance = 4.2351647363e-22 LI =         71 log2(LI) =   6; time =  0 seconds 
n =  72 distance = 2.1175823681e-22 LI =         72 log2(LI) =   6; time =  0 seconds 
n =  73 distance = 1.0587911841e-22 LI =         73 log2(LI) =   6; time =  0 seconds 
n =  74 distance = 5.2939559203e-23 LI =         74 log2(LI) =   6; time =  0 seconds 
n =  75 distance = 2.6469779602e-23 LI =         75 log2(LI) =   6; time =  0 seconds 
n =  76 distance = 1.3234889801e-23 LI =         76 log2(LI) =   6; time =  0 seconds 
n =  77 distance = 6.6174449004e-24 LI =         77 log2(LI) =   6; time =  0 seconds 
n =  78 distance = 3.3087224502e-24 LI =         78 log2(LI) =   6; time =  0 seconds 
n =  79 distance = 1.6543612251e-24 LI =         79 log2(LI) =   6; time =  0 seconds 
n =  80 distance = 8.2718061255e-25 LI =         80 log2(LI) =   6; time =  0 seconds 
n =  81 distance = 4.1359030628e-25 LI =         81 log2(LI) =   6; time =  0 seconds 
n =  82 distance = 2.0679515314e-25 LI =         82 log2(LI) =   6; time =  0 seconds 
n =  83 distance = 1.0339757657e-25 LI =         83 log2(LI) =   6; time =  0 seconds 
n =  84 distance = 5.1698788285e-26 LI =         84 log2(LI) =   6; time =  0 seconds 
n =  85 distance = 2.5849394142e-26 LI =         85 log2(LI) =   6; time =  0 seconds 
n =  86 distance = 1.2924697071e-26 LI =         86 log2(LI) =   6; time =  0 seconds 
n =  87 distance = 6.4623485356e-27 LI =         87 log2(LI) =   6; time =  0 seconds 
n =  88 distance = 3.2311742678e-27 LI =         88 log2(LI) =   6; time =  0 seconds 
n =  89 distance = 1.6155871339e-27 LI =         89 log2(LI) =   6; time =  0 seconds 
n =  90 distance = 8.0779356695e-28 LI =         90 log2(LI) =   6; time =  0 seconds 
n =  91 distance = 4.0389678347e-28 LI =         91 log2(LI) =   7; time =  0 seconds 
n =  92 distance = 2.0194839174e-28 LI =         92 log2(LI) =   7; time =  0 seconds 
n =  93 distance = 1.0097419587e-28 LI =         93 log2(LI) =   7; time =  0 seconds 
n =  94 distance = 5.0487097934e-29 LI =         94 log2(LI) =   7; time =  0 seconds 
n =  95 distance = 2.5243548967e-29 LI =         95 log2(LI) =   7; time =  0 seconds 
n =  96 distance = 1.2621774484e-29 LI =         96 log2(LI) =   7; time =  0 seconds 
n =  97 distance = 6.3108872418e-30 LI =         97 log2(LI) =   7; time =  0 seconds 
n =  98 distance = 3.1554436209e-30 LI =         98 log2(LI) =   7; time =  0 seconds 
n =  99 distance = 1.5777218104e-30 LI =         99 log2(LI) =   7; time =  0 seconds 
n = 100 distance = 7.8886090522e-31 LI =        100 log2(LI) =   7; time =  0 seconds 
n = 101 distance = 3.9443045261e-31 LI =        101 log2(LI) =   7; time =  0 seconds 
n = 102 distance = 1.9721522631e-31 LI =        102 log2(LI) =   7; time =  0 seconds 
n = 103 distance = 9.8607613153e-32 LI =        103 log2(LI) =   7; time =  0 seconds 
n = 104 distance = 4.9303806576e-32 LI =        104 log2(LI) =   7; time =  0 seconds 
n = 105 distance = 2.4651903288e-32 LI =        105 log2(LI) =   7; time =  0 seconds 
n = 106 distance = 1.2325951644e-32 LI =        106 log2(LI) =   7; time =  0 seconds 
n = 107 distance = 6.1629758220e-33 LI =        107 log2(LI) =   7; time =  0 seconds 
n = 108 distance = 3.0814879110e-33 LI =        108 log2(LI) =   7; time =  0 seconds 
n = 109 distance = 1.5407439555e-33 LI =        109 log2(LI) =   7; time =  0 seconds 
n = 110 distance = 7.7037197775e-34 LI =        110 log2(LI) =   7; time =  0 seconds 
n = 111 distance = 3.8518598888e-34 LI =        111 log2(LI) =   7; time =  0 seconds 
n = 112 distance = 1.9259299444e-34 LI =        112 log2(LI) =   7; time =  0 seconds 
n = 113 distance = 9.6296497219e-35 LI =        113 log2(LI) =   7; time =  0 seconds 
n = 114 distance = 4.8148248610e-35 LI =        114 log2(LI) =   7; time =  0 seconds 
n = 115 distance = 2.4074124305e-35 LI =        115 log2(LI) =   7; time =  0 seconds 
n = 116 distance = 1.2037062152e-35 LI =        116 log2(LI) =   7; time =  0 seconds 
n = 117 distance = 6.0185310762e-36 LI =        117 log2(LI) =   7; time =  0 seconds 
n = 118 distance = 3.0092655381e-36 LI =        118 log2(LI) =   7; time =  0 seconds 
n = 119 distance = 1.5046327691e-36 LI =        119 log2(LI) =   7; time =  0 seconds 
n = 120 distance = 7.5231638453e-37 LI =        120 log2(LI) =   7; time =  0 seconds 
n = 121 distance = 3.7615819226e-37 LI =        121 log2(LI) =   7; time =  0 seconds 
n = 122 distance = 1.8807909613e-37 LI =        122 log2(LI) =   7; time =  0 seconds 
n = 123 distance = 9.4039548066e-38 LI =        123 log2(LI) =   7; time =  0 seconds 
n = 124 distance = 4.7019774033e-38 LI =        124 log2(LI) =   7; time =  0 seconds 
n = 125 distance = 2.3509887016e-38 LI =        125 log2(LI) =   7; time =  0 seconds 
n = 126 distance = 1.1754943508e-38 LI =        126 log2(LI) =   7; time =  0 seconds 
n = 127 distance = 5.8774717541e-39 LI =        127 log2(LI) =   7; time =  0 seconds

Parabolic case :

Using MPFR-3.0.0-p8 with GMP-4.3.2 with precision = 100 bits and Escape Radius = 2.000000 
n =   1 distance = 5.0000000000e-01 LI =           3 log2(LI) =   2; time =     0 seconds 
n =   2 distance = 2.5000000000e-01 LI =           5 log2(LI) =   2; time =     0 seconds 
n =   3 distance = 1.2500000000e-01 LI =          10 log2(LI) =   3; time =     0 seconds 
n =   4 distance = 6.2500000000e-02 LI =          19 log2(LI) =   4; time =     0 seconds 
n =   5 distance = 3.1250000000e-02 LI =          35 log2(LI) =   5; time =     0 seconds 
n =   6 distance = 1.5625000000e-02 LI =          68 log2(LI) =   6; time =     0 seconds 
n =   7 distance = 7.8125000000e-03 LI =         133 log2(LI) =   7; time =     0 seconds 
n =   8 distance = 3.9062500000e-03 LI =         261 log2(LI) =   8; time =     0 seconds 
n =   9 distance = 1.9531250000e-03 LI =         518 log2(LI) =   9; time =     0 seconds 
n =  10 distance = 9.7656250000e-04 LI =        1031 log2(LI) =  10; time =     0 seconds 
n =  11 distance = 4.8828125000e-04 LI =        2055 log2(LI) =  11; time =     0 seconds 
n =  12 distance = 2.4414062500e-04 LI =        4104 log2(LI) =  12; time =     0 seconds 
n =  13 distance = 1.2207031250e-04 LI =        8201 log2(LI) =  13; time =     0 seconds 
n =  14 distance = 6.1035156250e-05 LI =       16394 log2(LI) =  14; time =     0 seconds 
n =  15 distance = 3.0517578125e-05 LI =       32778 log2(LI) =  15; time =     0 seconds 
n =  16 distance = 1.5258789062e-05 LI =       65547 log2(LI) =  16; time =     0 seconds 
n =  17 distance = 7.6293945312e-06 LI =      131084 log2(LI) =  17; time =     0 seconds 
n =  18 distance = 3.8146972656e-06 LI =      262156 log2(LI) =  18; time =     0 seconds 
n =  19 distance = 1.9073486328e-06 LI =      524301 log2(LI) =  19; time =     0 seconds 
n =  20 distance = 9.5367431641e-07 LI =     1048590 log2(LI) =  20; time =     1 seconds 
n =  21 distance = 4.7683715820e-07 LI =     2097166 log2(LI) =  21; time =     0 seconds 
n =  22 distance = 2.3841857910e-07 LI =     4194319 log2(LI) =  22; time =     2 seconds 
n =  23 distance = 1.1920928955e-07 LI =     8388624 log2(LI) =  23; time =     2 seconds 
n =  24 distance = 5.9604644775e-08 LI =    16777232 log2(LI) =  24; time =     6 seconds 
n =  25 distance = 2.9802322388e-08 LI =    33554449 log2(LI) =  25; time =    11 seconds 
n =  26 distance = 1.4901161194e-08 LI =    67108882 log2(LI) =  26; time =    21 seconds 
n =  27 distance = 7.4505805969e-09 LI =   134217747 log2(LI) =  27; time =    42 seconds 
n =  28 distance = 3.7252902985e-09 LI =   268435475 log2(LI) =  28; time =    87 seconds 
n =  29 distance = 1.8626451492e-09 LI =   536870932 log2(LI) =  29; time =   175 seconds 
n =  30 distance = 9.3132257462e-10 LI =  1073741845 log2(LI) =  30; time =   351 seconds 
n =  31 distance = 4.6566128731e-10 LI =  2147483669 log2(LI) =  31; time =   698 seconds 
n =  32 distance = 2.3283064365e-10 LI =  4294967318 log2(LI) =  32; time =  1386 seconds 
n =  33 distance = 1.1641532183e-10 LI =  8589934615 log2(LI) =  33; time =  2714 seconds 
n =  34 distance = 5.8207660913e-11 LI = 17179869207 log2(LI) =  34; time =  5595 seconds 
n =  35 distance = 2.9103830457e-11 LI = 34359738392 log2(LI) =  35; time = 11175 seconds 
n =  36 distance = 1.4551915228e-11 LI = 68719476762 log2(LI) =  36; time = 22081 seconds

Analysis

Maximal n in hyperbolic case $n=n_{h}$ is almost the same as the precision of the significand ^[31]

 $n_{h}=precision-1$

In parabolic case maximal $n=n_{p}$ is

 $n_{p}={\frac {precision}{2}}$

Last Iteration ( escape time = iteration fro which abs(zn) > ER ) is : in hyperbolic case equal to n :

 $LI_{h}=n$

in parabolic case equal to 2^n :

 $LI_{p}=2^{n}$

Time of computations is proportional to number of iterations. In hyperbolic case is is short. In parabolic case grows quickly as number of iterations.

 $time[seconds]=6*2^{n-24}$

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

Why programs fails ?

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); // = 1.2207031250e-04 = 0,00012207

It is greater than machine epsilon:^[35]

distance > FLT_EPSILON // = pow(2, -24) = 5.96e-08 = 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 ?

Why MPFR / GMP is slower than standard library ?

Why parabolic dynamics is so weak ?

Parabolic dynamics

numerical problems

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

Example ^[37]

Zoom

manipulating numbers below $10^{-308}$ 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 two-dimensional pixel

/*
 precision  based on pixel spacing 
 code by Claude Heiland-Allen
 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 Heiland-Allen

view = " cre cim radius"
view="-0.75 0 1.5"
view="0.275336142511115 6.75982538465039e-3 0.666e-5" 
view="-0.16323108442468427133 1.03436384057374316916 1e-5"

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, 1.0e-15, 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

Time series of the Tent map for the parameter m=2.0 which shows numerical error

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/is-this-characteristic-of-tent-map-usually-observed

 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/how-to-generate-random-float-number-in-c


*/
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

References

Do Numerical Orbits of Chaotic Dynamical Processes Represent True Orbits?

[1] thoverflow question : /rounding-errors-in-images-of-julia-sets

[2] ChaosAndFractals by Mark McClure. 2.11 A few notes on computation

[3] t of series at gmane.org discussion

[4] th.stackexchange questions : levins-u-transformation

[5] th.stackexchange questions : accelerating-convergence-of-a-sequence

[6] sl-1.0 : Series Acceleration

[7] sl manual : Series-Acceleration

[8] Fractal Forum : series-acceleration

[9] wolframalpha series calculator

[10] Fractalshades documentation

[11] What you never wanted to know about floating point but will be forced to find out by volkerschatz

[12] stackoverflow : Dealing with lack of floating point precision in OpenCL particle system

[13] Heavy computing with GLSL – Part 5: Emulated quadruple precision by Henry Thasler

[14] w-many-floating-point-numbers-are-in-the-interval [0,1 ? by Daniel Lemire]

[15] Arb by Fredrik Johansson

[16] Computing π with Chudnovsky and GMP by Beej Jorgensen

[17] [:w:Extended precision: Extended precision in wikipedia]

[18] [:w:GNU MPFR|GNU MPFR in wikipedia]

[19] Significant Digits and Rounding Copyright © 2003–2014 by Stan Brown

[20] thforum : Rules for Significant Figures and Decimal Places

[21] Roundoff Error Robert P. Munafo, 1996 Dec 3.

[22] wikipedia : Shadowing_lemma

[23] Wolf Jung's test for precision of drawing parameter rays

[24] One can find it using program Mandel by Wolf Jung using Ray to point menu position ( or Y key)

[25] Dynamics in one complex variable: introductory lectures version of 9-5-91 Appendix G by John W. Milnor

[26] Parabolic Julia Sets are Polynomial Time Computable Mark Braverman

[27] Roundoff Error by Robert P. Munafo, 1996 Dec 3.

[28] th.stackexchange questions : what-is-the-shape-of-parabolic-critical-orbit

[29] Parabolic Julia Sets are Polynomial Time Computable by Mark Braverman

[30] ractalforums : numerical-problem-with-bailout-test

[31] wikipedia :Floating point , EEE_754

[32] wikipedia : Significant figures

[33] wikipedia : Loss of significance

[34] IEEE Arithmetic from Numerical Computation Guide by Oracle

[35] Machine epsilon

[36] Dyskusja po polsku na pl.comp.os.linux.programowanie

[37] Precision and mandel zoom using DEM/M

[38] Fractalshades-doc : math

[39] reenigne blog : arbitrary precision mandelbrot sets

[40] z : Bignum by Michael Condron

[41] ractint : Arbitrary Precision and Deep Zooming

[42] spro documentation : parmparm

[43] Pixel density

[44] Automatic Math-Precision Robert P. Munafo, 1993 Apr 15.

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