Fractals/Mathematics/Numerical

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Contents

ConvergenceEdit

" ... 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) [1]

Above problem you can find in parabolic Julia sets.

" your sequence was generated as ... the output of a fixed point iteration  . ... The Levin transformation[2] ... uses forward differences successively to remove error terms of an alternating series."[3]

Examples:

  • GSL library[4] [5]
  • "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 )[6]
  • series calulator [7]

PrecisionEdit

How to deal with lack of floating point precision:[8]

  • implement higher precision arithmetic than your hardware natively supports
    • software emulation ( emulating a double with two floats, fixed points numbers , ...)[9]
  • use algorithms that are more numerically stable


Libraries :

  • Arb by Fredrik Johansson[10]

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 bit have 2 states. One digit of decimal number needs aproximately 3.4 bits. [11]There are 10 decimal numbers, 3 bits have 8 states which is not enoughl, 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[12]
mpfr_t[13] 142 41
mpfr_t 1000 294

External raysEdit

Parameter raysEdit

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. [14]

Dynamic raysEdit

number typeEdit

angle precisionEdit

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 :[15]

 


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(za-zb);
(%o3)                                 1.0

Escaping testEdit

This test was introduced by by John Milnor [16]. See also analysis by Mark Braverman [17] and roundoff error by Robert P. Munafo[18]

 
Julia set z+z^5. Image and src code

Comment by Mark McClure : [19] " 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 "

CasesEdit

Nonparabolic caseEdit

Lets take simple hyperbolic case where parameter c is :

 

Here repelling fixed point z_f is :

 


Parabolic caseEdit

Lets take simple parabolic case where parameter c is : [20]

 

Here parabolic fixed point z_f is :

 

TestEdit

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 )

ProgramsEdit

See FractalForum for evaldraw script[21]


ResultsEdit

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 

AnalysisEdit

Maximal n in hyperbolic case   is almost the same as the precision of the significand [22]

 

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&AEdit

Why programs fails ?Edit

Cancellation of significant digits[23] and loss of significance (LOS).[24][25]

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 then machine epsilon[26] :


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[27]

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

Finding roots of equationsEdit

Methods :


Programs:

square root of Negative or complex numberEdit

If S is negative real numnber :

 
 

then its principal square root is

 

If S is a complex number : S = a+bi where a and b are real and b ≠ 0,


then its principal square root ( = the root with the non-negative real part) is :

 

where :

 

is the absolute value (modulus) of S.

This can be verified by squaring the root.[28][29]

Distance estimationEdit

Example [30]

ZoomEdit

What precision do I need for zoom ? [31][32][33][34]

  • Pixel density [35]
  • 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 :

#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;

}

See alsoEdit

ReferencesEdit

  1. limit of series at gmane.org discussion
  2. math.stackexchange questions : levins-u-transformation
  3. math.stackexchange questions : accelerating-convergence-of-a-sequence
  4. gsl-1.0 : Series Acceleration
  5. gsl manual  : Series-Acceleration
  6. Fractal Forum : series-acceleration
  7. wolframalpha series calculator
  8. stackoverflow : Dealing with lack of floating point precision in OpenCL particle system
  9. Heavy computing with GLSL – Part 5: Emulated quadruple precision by Henry Thasler
  10. Arb by Fredrik Johansson
  11. Computing π with Chudnovsky and GMP by Beej Jorgensen
  12. [:w:Extended precision: Extended precision in wikipedia]
  13. [:w:GNU MPFR|GNU MPFR in wikipedia]
  14. Wolf Jung's test for precision of drawing parameter rays
  15. One can find it using program Mandel by Wolf Jung using Ray to point menu position ( or Y key)
  16. Dynamics in one complex variable: introductory lectures version of 9-5-91 Appendix G by John W. Milnor
  17. Parabolic Julia Sets are Polynomial Time Computable Mark Braverman
  18. Roundoff Error by Robert P. Munafo, 1996 Dec 3.
  19. math.stackexchange questions : what-is-the-shape-of-parabolic-critical-orbit
  20. Parabolic Julia Sets are Polynomial Time Computable by Mark Braverman
  21. fractalforums : numerical-problem-with-bailout-test
  22. wikipedia :Floating point , EEE_754
  23. wikipedia : Significant figures
  24. wikipedia : Loss of significance
  25. IEEE Arithmetic from Numerical Computation Guide by Oracle
  26. Machine epsilon
  27. Dyskusja po polsku na pl.comp.os.linux.programowanie
  28. Abramowitz, Miltonn; Stegun, Irene A. (1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Courier Dover Publications. p. 17. ISBN 0-486-61272-4. http://books.google.com/books?id=MtU8uP7XMvoC. , Section 3.7.26, p. 17
  29. Cooke, Roger (2008). Classical algebra: its nature, origins, and uses. John Wiley and Sons. p. 59. ISBN 0-470-25952-3. http://books.google.com/books?id=lUcTsYopfhkC. , Extract: page 59
  30. Precision and mandel zoom using DEM/M
  31. reenigne blog : arbitrary precision mandelbrot sets
  32. hpdz : Bignum by Michael Condron
  33. fractint : Arbitrary Precision and Deep Zooming
  34. chaospro documentation : parmparm
  35. Pixel density