Jacobi polynomials and recursive functions in arb4j's expression compiler

[Stephen Crowley]

I know no one cares, but Im close to finished with the Jacobi polynomial implementation in arb4j . There is a little more work to be done on the expression compiler, but it's quite functional now.

The following is generated by my code at https://github.com/crowlogic/arb4j/blob/master/src/main/java/arb/functions/polynomials/orthogonal/JacobiPolynomialSequence.java

I currently have the code referring to the decompiled versions of the class files generated by https://github.com/crowlogic/arb4j/blob/master/src/main/java/arb/expressions/Expression.java as

a debugging step.... the decompiled versions have very little modification and are currently there to facilitate debugging

P(1)=0.5*x

P(2)=0.75*x^2 - 0.375

P(3)=1.25*x^3 - 0.9375*x

P(4)=2.1875*x^4 - 2.1875*x^2 + 0.2734375

P(5)=3.9375*x^5 - 4.921875*x^3 + 1.23046875*x

P(6)=7.21875*x^6 - 10.828125*x^4 + 4.060546875*x^2 - 0.2255859375

P(7)=13.40625*x^7 - 23.4609375*x^5 + 11.73046875*x^3 - 1.46630859375*x

P(8)=25.13671875*x^8 - 50.2734375*x^6 + 31.4208984375*x^4 - 6.2841796875*x^2 + 0.196380615234375

The code to print the RealPolynomial such that it looks like a polynomial rather than just an array is at

https://github.com/crowlogic/arb4j/blob/master/src/main/java/arb/RealPolynomial.java

I will modify it soon to support writing the exponent in UTF chars so as to avoid the ascii representation of x^p

@Override

public String toString()

{

if (getLength() == 0)

{

return "∅";

}

if (getLength() == 1)

{

return get(0).toString();

}

StringBuilder builder = new StringBuilder();

for (int i = getLength() - 1; i >= 0; --i)

{

Real xi = get(i);

if (!xi.isZero())

{

if (i < getLength() - 1)

{

builder.append(xi.sign() >= 0 ? " + " : " ");

}

if (!xi.isOne())

{

builder.append(xi);

}

if (i > 0)

{

builder.append("*x");

if (i > 1)

{

builder.append("^" + i);

}

}

}

}

return (builder.toString() + (remainder != null ? " with remainder " + remainder : "")).replaceAll("-", "- ");

}

the recursions are a different form than how they are usually presented, this way avoids repeated calculations

The equivalent code that would have to be manually written if it were not for the expression compiler , just to evaluate the P function , is shown here

https://github.com/crowlogic/arb4j/blob/master/src/main/java/arb/functions/JacobiPolynomialPrototype.java

quite long, verbose, tedious and error-prone

[Stephen Crowley]

The JacobiPolynomialPrototype link should have been

https://github.com/crowlogic/arb4j/blob/master/src/main/java/arb/functions/P.java

[Stephen Crowley]

I think this syntax for the RealPolynomial java interface for arb_t is pretty good but it would be even better to have a way of reliably converting exact fractions.. or maybe I should just focus on on making a RealRationalPolynomial interface for fmpq_poly_t and implement the recursions directly with the real rational polynomial type.

Anyone have any opionions?

https://github.com/crowlogic/arb4j/blob/master/src/main/java/arb/functions/polynomials/orthogonal/JacobiPolynomialSequence.java

https://github.com/crowlogic/arb4j/blob/master/src/main/java/arb/RealPolynomial.java

Here's the first 20 Jacobi polynomials with α=β=-½

P(0,x)=1

P(1,x)=0.5x

P(2,x)=0.75x² - 0.375

P(3,x)=1.25x³ - 0.9375x

P(4,x)=2.1875x⁴ - 2.1875x² + 0.2734375

P(5,x)=3.9375x⁵ - 4.921875x³ + 1.23046875x

P(6,x)=7.21875x⁶ - 10.828125x⁴ + 4.060546875x² - 0.2255859375

P(7,x)=13.40625x⁷ - 23.4609375x⁵ + 11.73046875x³ - 1.46630859375x

P(8,x)=25.13671875x⁸ - 50.2734375x⁶ + 31.4208984375x⁴ - 6.2841796875x² + 0.196380615234375

P(9,x)=47.48046875x⁹ - 106.8310546875x⁷ + 80.123291015625x⁵ - 22.2564697265625x³ + 1.6692352294921875x

P(10,x)=90.212890625x¹⁰ - 225.5322265625x⁸ + 197.3406982421875x⁶ - 70.47882080078125x⁴ + 8.80985260009765625x² - 0.176197052001953125

P(11,x)=172.224609375x¹¹ - 473.61767578125x⁹ + 473.61767578125x⁷ - 207.207733154296875x⁵ + 37.00138092041015625x³ - 1.8500690460205078125x

P(12,x)=330.09716796875x¹² - 990.29150390625x¹⁰ + 1114.07794189453125x⁸ - 577.6700439453125x⁶ + 135.3914165496826171875x⁴ - 11.6049785614013671875x² + 0.1611802577972412109375

P(13,x)=634.80224609375x¹³ - 2063.1072998046875x¹¹ + 2578.884124755859375x⁹ - 1547.330474853515625x⁷ + 451.304721832275390625x⁵ - 56.413090229034423828125x³ + 2.01475322246551513671875x

P(14,x)=1224.261474609375x¹⁴ - 4284.9151611328125x¹² + 5891.7583465576171875x¹⁰ - 4017.10796356201171875x⁸ + 1405.9877872467041015625x⁶ - 234.33129787445068359375x⁴ + 14.645706117153167724609375x² - 0.1494459807872772216796875

P(15,x)=2366.905517578125x¹⁵ - 8875.89569091796875x¹³ + 13313.843536376953125x¹¹ - 10170.297145843505859375x⁹ + 4160.5761051177978515625x⁷ - 873.720982074737548828125x⁵ + 80.9000909328460693359375x³ - 2.16696672141551971435546875x

P(16,x)=4585.8794403076171875x¹⁶ - 18343.51776123046875x¹⁴ + 29808.21636199951171875x¹² - 25222.33692169189453125x¹⁰ + 11822.9704320430755615234375x⁸ - 3009.483382701873779296875x⁶ + 376.185422837734222412109375x⁴ - 17.913591563701629638671875x² + 0.1399499340914189815521240234375

P(17,x)=8902.0012664794921875x¹⁷ - 37833.505382537841796875x¹⁵ + 66208.63441944122314453125x¹³ - 61479.446246623992919921875x¹¹ + 32513.168688118457794189453125x⁹ - 9753.9506064355373382568359375x⁷ + 1551.764869205653667449951171875x⁵ - 110.8403478004038333892822265625x³ + 2.30917391250841319561004638671875x

P(18,x)=17309.44690704345703125x¹⁸ - 77892.511081695556640625x¹⁶ + 146048.458278179168701171875x¹⁴ - 147671.2189257144927978515625x¹² + 87020.5397240817546844482421875x¹⁰ - 30122.49451987445354461669921875x⁸ + 5857.1517121978104114532470703125x⁶ - 570.50179014913737773895263671875x⁴ + 21.393817130592651665210723876953125x² - 0.1320605995715595781803131103515625

P(19,x)=33707.87029266357421875x¹⁹ - 160112.3838901519775390625x¹⁷ + 320224.767780303955078125x¹⁵ - 350245.83975970745086669921875x¹³ + 227659.7958438098430633544921875x¹¹ - 89437.776938639581203460693359375x⁹ + 20639.486985839903354644775390625x⁷ - 2579.935873229987919330596923828125x⁵ + 146.587265524431131780147552490234375x³ - 2.44312109207385219633579254150390625x

[Jeffrey Sarnoff]

A distinct RationalPolynomial type would be cleaner and more easily optimized.

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