Converting double to BigDecimal in Java

Question

I wrote a Java program that calculates values for the Riemann Zeta Function. Inside the program, I made a library to calculate necessary complex functions such as atan, cos, etc. Everything inside both programs is accessed through the double and BigDecimal data types. This creates major issues when evaluating large values for the Zeta function.

The numerical approximation for the Zeta function references

Directly evaluating this approximation at high values creates issues when s has a large complex form, such as s = (230+30i). I am very grateful to get information about this here. The evaluation of S2.minus(S1) creates errors because I wrote something wrong in the adaptiveQuad method.

As an example, Zeta(2+3i) through this program generates

Calculation of the Riemann Zeta Function in the form Zeta(s) = a + ib.  Enter the value of [a] inside the Riemann Zeta Function: 2 Enter the value of [b] inside the Riemann Zeta Function: 3 The value for Zeta(s) is 7.980219851133409E-1 - 1.137443081631288E-1*i Total time taken is 0.469 seconds. 

Which is correct.

Zeta(100+0i) generates

Calculation of the Riemann Zeta Function in the form Zeta(s) = a + ib.  Enter the value of [a] inside the Riemann Zeta Function: 100 Enter the value of [b] inside the Riemann Zeta Function: 0 The value for Zeta(s) is 1.000000000153236E0 Total time taken is 0.672 seconds. 

Which is also correct as compared to Wolfram. The problem is due to something inside the method labelled adaptiveQuad.

Zeta(230+30i) generates

Calculation of the Riemann Zeta Function in the form Zeta(s) = a + ib.  Enter the value of [a] inside the Riemann Zeta Function: 230 Enter the value of [b] inside the Riemann Zeta Function: 30 The value for Zeta(s) is 0.999999999999093108519845391615339162047254997503854254342793916541606842461539820124897870147977114468145672577664412128509813042591501204781683860384769321084473925620572315416715721728082468412672467499199310913504362891199180150973087384370909918493750428733837552915328069343498987460727711606978118652477860450744628906250 - 38.005428584222228490409289204403133867487950535704812764806874887805043029499897666636162309572126423385487374863788363786029170239477119910868455777891701471328505006916099918492113970510619110472506796418206225648616641319533972054228283869713393805956289770456519729094756021581247296126093715429306030273437500E-15*i Total time taken is 1.746 seconds. 

The imaginary part is a bit off as compared to Wolfram.

The algorithm to evaluate the integral is known as Adaptive Quadrature and a double Java implementation is found here. The adaptive quad method applies the following

// adaptive quadrature public static double adaptive(double a, double b) {     double h = b - a;     double c = (a + b) / 2.0;     double d = (a + c) / 2.0;     double e = (b + c) / 2.0;     double Q1 = h/6  * (f(a) + 4*f(c) + f(b));     double Q2 = h/12 * (f(a) + 4*f(d) + 2*f(c) + 4*f(e) + f(b));     if (Math.abs(Q2 - Q1) <= EPSILON)         return Q2 + (Q2 - Q1) / 15;     else         return adaptive(a, c) + adaptive(c, b); } 

Here is my fourth attempt at writing the program

/************************************************************************** ** **    Abel-Plana Formula for the Zeta Function ** ************************************************************************** **    Axion004 **    08/16/2015 ** **    This program computes the value for Zeta(z) using a definite integral **    approximation through the Abel-Plana formula. The Abel-Plana formula **    can be shown to approximate the value for Zeta(s) through a definite **    integral. The integral approximation is handled through the Composite **    Simpson's Rule known as Adaptive Quadrature. **************************************************************************/  import java.util.*; import java.math.*;   public class AbelMain5 extends Complex {     private static MathContext MC = new MathContext(512,             RoundingMode.HALF_EVEN);     public static void main(String[] args) {         AbelMain();     }      // Main method     public static void AbelMain() {         double re = 0, im = 0;         double start, stop, totalTime;         Scanner scan = new Scanner(System.in);         System.out.println("Calculation of the Riemann Zeta " +                 "Function in the form Zeta(s) = a + ib.");         System.out.println();         System.out.print("Enter the value of [a] inside the Riemann Zeta " +                 "Function: ");         try {                 re = scan.nextDouble();         }         catch (Exception e) {            System.out.println("Please enter a valid number for a.");         }         System.out.print("Enter the value of [b] inside the Riemann Zeta " +                 "Function: ");         try {                 im = scan.nextDouble();         }         catch (Exception e) {            System.out.println("Please enter a valid number for b.");         }         start = System.currentTimeMillis();         Complex z = new Complex(new BigDecimal(re), new BigDecimal(im));         System.out.println("The value for Zeta(s) is " + AbelPlana(z));         stop = System.currentTimeMillis();         totalTime = (double) (stop-start) / 1000.0;         System.out.println("Total time taken is " + totalTime + " seconds.");     }      /**      * The definite integral for Zeta(z) in the Abel-Plana formula.          * <br> Numerator = Sin(z * arctan(t))          * <br> Denominator = (1 + t^2)^(z/2) * (e^(2*pi*t) - 1)      * @param t - the value of t passed into the integrand.          * @param z - The complex value of z = a + i*b      * @return the value of the complex function.     */     public static Complex f(double t, Complex z) {         Complex num = (z.multiply(Math.atan(t))).sin();         Complex D1 = new Complex(1 + t*t).pow(z.divide(TWO));         Complex D2 = new Complex(Math.pow(Math.E, 2.0*Math.PI*t) - 1.0);         Complex den = D1.multiply(D2);         return num.divide(den, MC);     }      /**      * Adaptive quadrature - See http://www.mathworks.com/moler/quad.pdf      * @param a - the lower bound of integration.          * @param b - the upper bound of integration.          * @param z - The complex value of z = a + i*b      * @return the approximate numerical value of the integral.     */     public static Complex adaptiveQuad(double a, double b, Complex z) {         double EPSILON = 1E-10;         double step = b - a;         double c = (a + b) / 2.0;         double d = (a + c) / 2.0;         double e = (b + c) / 2.0;          Complex S1 = (f(a, z).add(f(c, z).multiply(FOUR)).add(f(b, z))).                 multiply(step / 6.0);         Complex S2 = (f(a, z).add(f(d, z).multiply(FOUR)).add(f(c, z).multiply                 (TWO)).add(f(e, z).multiply(FOUR)).add(f(b, z))).multiply                 (step / 12.0);         Complex result = (S2.subtract(S1)).divide(FIFTEEN, MC);          if(S2.subtract(S1).mod() <= EPSILON)             return S2.add(result);         else             return adaptiveQuad(a, c, z).add(adaptiveQuad(c, b, z));     }      /**      * The definite integral for Zeta(z) in the Abel-Plana formula.          * <br> value =  1/2 + 1/(z-1) + 2 * Integral          * @param z - The complex value of z = a + i*b      * @return the value of Zeta(z) through value and the          * quadrature approximation.     */     public static Complex AbelPlana(Complex z) {         Complex C1 = ONEHALF.add(ONE.divide(z.subtract(ONE), MC));         Complex C2  =  TWO.multiply(adaptiveQuad(1E-16, 100.0, z));         if ( z.real().doubleValue() == 0 && z.imag().doubleValue() == 0)             return new Complex(0.0, 0.0);         else             return C1.add(C2);     }  } 

Complex numbers (BigDecimal)

/************************************************************************** ** **    Complex Numbers ** ************************************************************************** **    Axion004 **    08/20/2015 ** **    This class is necessary as a helper class for the calculation of **    imaginary numbers. The calculation of Zeta(z) inside AbelMain is in **    the form of z = a + i*b. **************************************************************************/  import java.math.BigDecimal; import java.math.MathContext; import java.text.DecimalFormat; import java.text.NumberFormat;  public class Complex extends Object{     private BigDecimal re;     private BigDecimal im;      /**         BigDecimal constant for zero     */     final static Complex ZERO = new Complex(BigDecimal.ZERO) ;      /**         BigDecimal constant for one half     */     final static Complex ONEHALF = new Complex(new BigDecimal(0.5));      /**         BigDecimal constant for one     */     final static Complex ONE = new Complex(BigDecimal.ONE);      /**         BigDecimal constant for two     */     final static Complex TWO = new Complex(new BigDecimal(2.0));      /**         BigDecimal constant for four     */     final static Complex FOUR = new Complex(new BigDecimal(4.0)) ;      /**         BigDecimal constant for fifteen     */     final static Complex FIFTEEN = new Complex(new BigDecimal(15.0)) ;      /**         Default constructor equivalent to zero     */     public Complex() {         re = BigDecimal.ZERO;         im = BigDecimal.ZERO;     }      /**         Constructor with real part only         @param x Real part, BigDecimal     */     public Complex(BigDecimal x) {         re = x;         im = BigDecimal.ZERO;     }      /**         Constructor with real part only         @param x Real part, double     */     public Complex(double x) {         re = new BigDecimal(x);         im = BigDecimal.ZERO;     }      /**         Constructor with real and imaginary parts in double format.         @param x Real part         @param y Imaginary part     */     public Complex(double x, double y) {         re= new BigDecimal(x);         im= new BigDecimal(y);     }      /**         Constructor for the complex number z = a + i*b         @param re Real part         @param im Imaginary part     */     public Complex (BigDecimal re, BigDecimal im) {         this.re = re;         this.im = im;     }      /**         Real part of the Complex number         @return Re[z] where z = a + i*b.     */     public BigDecimal real() {         return re;     }      /**         Imaginary part of the Complex number         @return Im[z] where z = a + i*b.     */     public BigDecimal imag() {         return im;     }      /**         Complex conjugate of the Complex number         in which the conjugate of z is z-bar.         @return z-bar where z = a + i*b and z-bar = a - i*b     */     public Complex conjugate() {         return new Complex(re, im.negate());     }      /**      * Returns the sum of this and the parameter.         @param augend the number to add        @param mc the context to use        @return this + augend      */     public Complex add(Complex augend,MathContext mc)     {         //(a+bi)+(c+di) = (a + c) + (b + d)i         return new Complex(             re.add(augend.re,mc),             im.add(augend.im,mc));     }      /**        Equivalent to add(augend, MathContext.UNLIMITED)        @param augend the number to add        @return this + augend      */     public Complex add(Complex augend)     {         return add(augend, MathContext.UNLIMITED);     }      /**         Addition of Complex number and a double.         @param d is the number to add.         @return z+d where z = a+i*b and d = double     */     public Complex add(double d){         BigDecimal augend = new BigDecimal(d);         return new Complex(this.re.add(augend, MathContext.UNLIMITED),                 this.im);     }      /**      * Returns the difference of this and the parameter.        @param subtrahend the number to subtract        @param mc the context to use        @return this - subtrahend      */     public Complex subtract(Complex subtrahend, MathContext mc)     {         //(a+bi)-(c+di) = (a - c) + (b - d)i         return new Complex(             re.subtract(subtrahend.re,mc),             im.subtract(subtrahend.im,mc));     }      /**      * Equivalent to subtract(subtrahend, MathContext.UNLIMITED)        @param subtrahend the number to subtract        @return this - subtrahend      */     public Complex subtract(Complex subtrahend)     {         return subtract(subtrahend,MathContext.UNLIMITED);     }      /**         Subtraction of Complex number and a double.         @param d is the number to subtract.         @return z-d where z = a+i*b and d = double     */     public Complex subtract(double d){         BigDecimal subtrahend = new BigDecimal(d);         return new Complex(this.re.subtract(subtrahend, MathContext.UNLIMITED),                 this.im);     }      /**      * Returns the product of this and the parameter.        @param multiplicand the number to multiply by        @param mc the context to use        @return this * multiplicand      */     public Complex multiply(Complex multiplicand, MathContext mc)     {         //(a+bi)(c+di) = (ac - bd) + (ad + bc)i         return new Complex(             re.multiply(multiplicand.re,mc).subtract(im.multiply                 (multiplicand.im,mc),mc),             re.multiply(multiplicand.im,mc).add(im.multiply                 (multiplicand.re,mc),mc));     }      /**        Equivalent to multiply(multiplicand, MathContext.UNLIMITED)        @param multiplicand the number to multiply by        @return this * multiplicand      */     public Complex multiply(Complex multiplicand)     {         return multiply(multiplicand,MathContext.UNLIMITED);     }      /**         Complex multiplication by a double.         @param d is the double to multiply by.         @return z*d where z = a+i*b and d = double     */     public Complex multiply(double d){         BigDecimal multiplicand = new BigDecimal(d);         return new Complex(this.re.multiply(multiplicand, MathContext.UNLIMITED)                 ,this.im.multiply(multiplicand, MathContext.UNLIMITED));     }      /**         Modulus of a Complex number or the distance from the origin in         * the polar coordinate plane.         @return |z| where z = a + i*b.     */     public double mod() {         if ( re.doubleValue() != 0.0 || im.doubleValue() != 0.0)             return Math.sqrt(re.multiply(re).add(im.multiply(im))                     .doubleValue());         else             return 0.0;     }      /**      * Modulus of a Complex number squared      * @param z = a + i*b      * @return |z|^2 where z = a + i*b     */     public double abs(Complex z) {         double doubleRe = re.doubleValue();         double doubleIm = im.doubleValue();         return doubleRe * doubleRe + doubleIm * doubleIm;     }      public Complex divide(Complex divisor)     {         return divide(divisor,MathContext.UNLIMITED);     }      /**         * The absolute value squared.         * @return The sum of the squares of real and imaginary parts.         * This is the square of Complex.abs() .         */     public BigDecimal norm()     {                 return re.multiply(re).add(im.multiply(im)) ;     }      /**         * The absolute value of a BigDecimal.         * @param mc amount of precision         * @return BigDecimal.abs()         */     public BigDecimal abs(MathContext mc)     {                 return BigDecimalMath.sqrt(norm(),mc) ;     }      /** The inverse of the the Complex number.           @param mc amount of precision           @return 1/this         */     public Complex inverse(MathContext mc)     {                 final BigDecimal hyp = norm() ;                 /* 1/(x+iy)= (x-iy)/(x^2+y^2 */                 return new Complex( re.divide(hyp,mc), im.divide(hyp,mc)                         .negate() ) ;     }      /** Divide through another BigComplex number.         @param oth the other complex number         @param mc amount of precision         @return this/other         */     public Complex divide(Complex oth, MathContext mc)     {                 /* implementation: (x+iy)/(a+ib)= (x+iy)* 1/(a+ib) */                 return multiply(oth.inverse(mc),mc) ;     }     /**         Division of Complex number by a double.         @param d is the double to divide         @return new Complex number z/d where z = a+i*b     */     public Complex divide(double d){         BigDecimal divisor = new BigDecimal(d);         return new Complex(this.re.divide(divisor, MathContext.UNLIMITED),                 this.im.divide(divisor, MathContext.UNLIMITED));     }      /**         Exponential of a complex number (z is unchanged).         <br> e^(a+i*b) = e^a * e^(i*b) = e^a * (cos(b) + i*sin(b))         @return exp(z) where z = a+i*b     */     public Complex exp () {         return new Complex(Math.exp(re.doubleValue()) * Math.cos(im.                 doubleValue()), Math.exp(re.doubleValue()) *                 Math.sin(im.doubleValue()));     }       /**         The Argument of a Complex number or the angle in radians         with respect to polar coordinates.         <br> Tan(theta) = b / a, theta = Arctan(b / a)         <br> a is the real part on the horizontal axis         <br> b is the imaginary part of the vertical axis         @return arg(z) where z = a+i*b.     */     public double arg() {         return Math.atan2(im.doubleValue(), re.doubleValue());     }      /**         The log or principal branch of a Complex number (z is unchanged).         <br> Log(a+i*b) = ln|a+i*b| + i*Arg(z) = ln(sqrt(a^2+b^2))         * + i*Arg(z) = ln (mod(z)) + i*Arctan(b/a)         @return log(z) where z = a+i*b     */     public Complex log() {         return new Complex(Math.log(this.mod()), this.arg());     }      /**         The square root of a Complex number (z is unchanged).         Returns the principal branch of the square root.         <br> z = e^(i*theta) = r*cos(theta) + i*r*sin(theta)         <br> r = sqrt(a^2+b^2)         <br> cos(theta) = a / r, sin(theta) = b / r         <br> By De Moivre's Theorem, sqrt(z) = sqrt(a+i*b) =         * e^(i*theta / 2) = r(cos(theta/2) + i*sin(theta/2))         @return sqrt(z) where z = a+i*b     */     public Complex sqrt() {         double r = this.mod();         double halfTheta = this.arg() / 2;         return new Complex(Math.sqrt(r) * Math.cos(halfTheta), Math.sqrt(r) *                 Math.sin(halfTheta));     }      /**         The real cosh function for Complex numbers.         <br> cosh(theta) = (e^(theta) + e^(-theta)) / 2         @return cosh(theta)     */     private double cosh(double theta) {         return (Math.exp(theta) + Math.exp(-theta)) / 2;     }      /**         The real sinh function for Complex numbers.         <br> sinh(theta) = (e^(theta) - e^(-theta)) / 2         @return sinh(theta)     */     private double sinh(double theta) {         return (Math.exp(theta) - Math.exp(-theta)) / 2;     }       /**         The sin function for the Complex number (z is unchanged).         <br> sin(a+i*b) = cosh(b)*sin(a) + i*(sinh(b)*cos(a))         @return sin(z) where z = a+i*b     */     public Complex sin() {         return new Complex(cosh(im.doubleValue()) * Math.sin(re.doubleValue()),                 sinh(im.doubleValue())* Math.cos(re.doubleValue()));     }      /**         The cos function for the Complex number (z is unchanged).         <br> cos(a +i*b) = cosh(b)*cos(a) + i*(-sinh(b)*sin(a))         @return cos(z) where z = a+i*b     */     public Complex cos() {         return new Complex(cosh(im.doubleValue()) * Math.cos(re.doubleValue()),                 -sinh(im.doubleValue()) * Math.sin(re.doubleValue()));     }      /**         The hyperbolic sin of the Complex number (z is unchanged).         <br> sinh(a+i*b) = sinh(a)*cos(b) + i*(cosh(a)*sin(b))         @return sinh(z) where z = a+i*b     */     public Complex sinh() {         return new Complex(sinh(re.doubleValue()) * Math.cos(im.doubleValue()),                 cosh(re.doubleValue()) * Math.sin(im.doubleValue()));     }      /**         The hyperbolic cosine of the Complex number (z is unchanged).         <br> cosh(a+i*b) = cosh(a)*cos(b) + i*(sinh(a)*sin(b))         @return cosh(z) where z = a+i*b     */     public Complex cosh() {         return new Complex(cosh(re.doubleValue()) *Math.cos(im.doubleValue()),                 sinh(re.doubleValue()) * Math.sin(im.doubleValue()));     }      /**         The tan of the Complex number (z is unchanged).         <br> tan (a+i*b) = sin(a+i*b) / cos(a+i*b)         @return tan(z) where z = a+i*b     */     public Complex tan() {         return (this.sin()).divide(this.cos());     }      /**         The arctan of the Complex number (z is unchanged).         <br> tan^(-1)(a+i*b) = 1/2 i*(log(1-i*(a+b*i))-log(1+i*(a+b*i))) =         <br> -1/2 i*(log(i*a - b+1)-log(-i*a + b+1))         @return arctan(z) where z = a+i*b     */     public Complex atan(){         Complex ima = new Complex(0.0,-1.0);    //multiply by negative i         Complex num = new Complex(this.re.doubleValue(),this.im.doubleValue()                 -1.0);         Complex den = new Complex(this.re.negate().doubleValue(),this.im                 .negate().doubleValue()-1.0);         Complex two = new Complex(2.0, 0.0);    // divide by 2         return ima.multiply(num.divide(den).log()).divide(two);     }      /**      * The Math.pow equivalent of two Complex numbers.      * @param z - the complex base in the form z = a + i*b      * @return z^y where z = a + i*b and y = c + i*d     */     public Complex pow(Complex z){         Complex a = z.multiply(this.log(), MathContext.UNLIMITED);         return a.exp();     }      /**      * The Math.pow equivalent of a Complex number to the power          * of a double.      * @param d - the double to be taken as the power.      * @return z^d where z = a + i*b and d = double     */      public Complex pow(double d){          Complex a=(this.log()).multiply(d);          return a.exp();      }      /**         Override the .toString() method to generate complex numbers, the         * string representation is now a literal Complex number.         @return a+i*b, a-i*b, a, or i*b as desired.     */     public String toString() {          NumberFormat formatter = new DecimalFormat();         formatter = new DecimalFormat("#.###############E0");          if (re.doubleValue() != 0.0 && im.doubleValue()  > 0.0) {             return formatter.format(re) + " + " + formatter.format(im)                     +"*i";         }         if (re.doubleValue() !=0.0 && im.doubleValue() < 0.0) {             return formatter.format(re) + " - "+ formatter.format(im.negate())                     + "*i";         }         if (im.doubleValue() == 0.0) {             return formatter.format(re);         }         if (re.doubleValue() == 0.0) {             return formatter.format(im) + "*i";         }         return formatter.format(re) + " + i*" + formatter.format(im);     } } 

I am reviewing the answer below.

One problem may be due to

Complex num = (z.multiply(Math.atan(t))).sin(); Complex D1 = new Complex(1 + t*t).pow(z.divide(TWO)); Complex D2 = new Complex(Math.pow(Math.E, 2.0*Math.PI*t) - 1.0); Complex den = D1.multiply(D2, MathContext.UNLIMITED); 

I am not applying BigDecimal.pow(BigDecimal). Although, I don't think this is the direct issue which causes the floating point arithmetic to create differences.

Edit: I tried a new integral approximation of the Zeta function. Ultimately, I will develop a new method to calculate BigDecimal.pow(BigDecimal).

Answer 1

Caveat I agree with all the comments in @laune's answer, but I get the impression you may wish to pursue this anyway. Make sure especially that you really do understand 1) and what that means for you - it's very easy to do a lot of heavy calculations to produce meaningless results.

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Arbitrary precision floating point functions in Java

To reiterate a little, I think your problem really is with the maths and numerical method you have chosen, but here's an implementation using the Apfloat library. I'd strongly urge you to use the ready made arbitrary precision library (or a similar one) as it avoids any need for you to "roll your own" arbitrary precision maths functions (such as pow, exp,sin, atan etc). You say

Ultimately, I will develop a new method to calculate BigDecimal.pow(BigDecimal)

It's really hard to get that right.

You need to watch the precision of your constants, too - note I use an Apfloat sample implementation to calculate PI to a large number (for some definition of large!) of sig figs. I am to some degree trusting that the Apfloat library uses suitably precise values for e in exponentiation - the source is available if you want to check.

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Different integral formulations to calculate zeta

You put up three different integration based methods in one of your edits:
The one labelled 25.5.12 is the one that you currently have in the question and (although that can be calculated at zero easily), it is hard to work with due to 2) in @laune's answer. I implemented 25.5.12 as integrand1() in the code - I urge you to plot it with range of t for different s = a + 0i and understand how it behaves. Or look at the plots in the zeta article on Wolfram's mathworld. The one labelled 25.5.11 I implemented via integrand2() and the code in the configuration I publish below.

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Code

While I'm a bit reluctant to post code that will no doubt find wrong results in some configurations due to all the things above - I have encoded what you are trying to do below, using arbitrary precision floating point objects for the variables.

If you want to change which formulation you use (e.g. from 25.5.11 to 25.5.12), you can change which integrand the wrapper function f() returns or, better yet, change adaptiveQuad to take in an arbitrary integrand method wrapped in a class with an interface... You will also have to alter the arithmetic in findZeta() if you want to use one of the other integral formulations.

Play with the constants at the start to your heart's content. I haven't tested lots of combinations, as I think the maths problems here override the programming ones.

I've left it set up to do 2+3i in about 2000 calls to the adaptive quadature method and match the first 15 or so digits of the Wolfram value.

I've tested it still works with PRECISION = 120l and EPSILON=1e-15. The program matches Wolfram alpha in the first 18 or so significant figures for the three test cases you provide. The last one (230+30i) takes a long time even on a fast computer - it calls the integrand fucntion some 100,000+ times. Note that I use 40 for the value of INFINITY in the integral - not very high - but higher values exhibit the problem 1) as already discussed...

N.B. This is not fast (you'll be measuring in minutes or hours, not seconds - but you only get really quick if you want to accept that 10^-15 ~= 10^-70 as most people would!!). It will give you some digits that match Wolfram Alpha ;) You might want to take PRECISION down to about 20, INFINITY to 10 and EPSILON to 1e-10 to verify a few results with small s first... I've left in some printing so it tells you every 100th time adaptiveQuad is called for comfort.

Reiteration However good your precision - it's not going to overcome the mathematical characteristics of the functions involved in this way of calculating zeta. I strongly doubt this is how Wolfram alpha does it, for instance. Look up series summation methods if you want more tractable methods.

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import java.io.PrintWriter; import org.apfloat.ApcomplexMath; import org.apfloat.Apcomplex; import org.apfloat.Apfloat; import org.apfloat.samples.Pi;  public class ZetaFinder {     //Number of sig figs accuracy.  Note that infinite should be reserved     private static long PRECISION = 40l;      // Convergence criterion for integration     static Apfloat EPSILON = new Apfloat("1e-15",PRECISION);      //Value of PI - enhanced using Apfloat library sample calculation of Pi in constructor,     //Fast enough that we don't need to hard code the value in.     //You could code hard value in for perf enhancement     static Apfloat PI = null; //new Apfloat("3.14159");       //Integration limits - I found too high a value for "infinity" causes integration     //to terminate on first iteration.  Plot the integrand to see why...     static Apfloat INFINITE_LIMIT = new Apfloat("40",PRECISION);     static Apfloat ZERO_LIMIT = new Apfloat("1e-16",PRECISION); //You can use zero for the 25.5.12      static Apfloat one = new Apfloat("1",PRECISION);     static Apfloat two = new Apfloat("2",PRECISION);     static Apfloat four = new Apfloat("4",PRECISION);     static Apfloat six = new Apfloat("6",PRECISION);     static Apfloat twelve = new Apfloat("12",PRECISION);     static Apfloat fifteen = new Apfloat("15",PRECISION);      static int counter = 0;      Apcomplex s = null;      public ZetaFinder(Apcomplex s)     {         this.s = s;         Pi.setOut(new PrintWriter(System.out, true));         Pi.setErr(new PrintWriter(System.err, true));         PI = (new Pi.RamanujanPiCalculator(PRECISION+10, 10)).execute(); //Get Pi to a higher precision than integer consts         System.out.println("Created a Zeta Finder based on Abel-Plana for s="+s.toString() + " using PI="+PI.toString());     }      public static void main(String[] args)     {         Apfloat re = new Apfloat("2", PRECISION);         Apfloat im = new Apfloat("3", PRECISION);         Apcomplex s = new Apcomplex(re,im);          ZetaFinder finder = new ZetaFinder(s);          System.out.println(finder.findZeta());     }      private Apcomplex findZeta()     {         Apcomplex retval = null;          //Method currently in question (a.k.a. 25.5.12)         //Apcomplex mult = ApcomplexMath.pow(two, this.s);         //Apcomplex firstterm = (ApcomplexMath.pow(two, (this.s.add(one.negate())))).divide(this.s.add(one.negate()));          //Easier integrand method (a.k.a. 25.5.11)         Apcomplex mult = two;         Apcomplex firstterm = (one.divide(two)).add(one.divide(this.s.add(one.negate())));          Apfloat limita = ZERO_LIMIT;//Apfloat.ZERO;         Apfloat limitb = INFINITE_LIMIT;         System.out.println("Trying to integrate between " + limita.toString() + " and " + limitb.toString());         Apcomplex integral = adaptiveQuad(limita, limitb);         retval = firstterm.add((mult.multiply(integral)));         return retval;     }      private Apcomplex adaptiveQuad(Apfloat a, Apfloat b) {         //if (counter % 100 == 0)         {             System.out.println("In here for the " + counter + "th time");         }         counter++;         Apfloat h = b.add(a.negate());         Apfloat c = (a.add(b)).divide(two);         Apfloat d = (a.add(c)).divide(two);         Apfloat e = (b.add(c)).divide(two);          Apcomplex Q1 = (h.divide(six)).multiply(f(a).add(four.multiply(f(c))).add(f(b)));         Apcomplex Q2 = (h.divide(twelve)).multiply(f(a).add(four.multiply(f(d))).add(two.multiply(f(c))).add(four.multiply(f(e))).add(f(b)));         if (ApcomplexMath.abs(Q2.add(Q1.negate())).compareTo(EPSILON) < 0)         {             System.out.println("Returning");             return Q2.add((Q2.add(Q1.negate())).divide(fifteen));         }         else         {             System.out.println("Recursing with intervals "+a+" to " + c + " and " + c + " to " +d);             return adaptiveQuad(a, c).add(adaptiveQuad(c, b));         }     }      private Apcomplex f(Apfloat x)     {         return integrand2(x);     }      /*      * Simple test integrand (z^2)      *       * Can test implementation by asserting that the adaptiveQuad      * with this function evaluates to z^3 / 3      */     private Apcomplex integrandTest(Apfloat t)     {         return ApcomplexMath.pow(t, two);     }      /*      * Abel-Plana formulation integrand      */     private Apcomplex integrand1(Apfloat t)     {         Apcomplex numerator = ApcomplexMath.sin(this.s.multiply(ApcomplexMath.atan(t)));         Apcomplex bottomlinefirstbr = one.add(ApcomplexMath.pow(t, two));         Apcomplex D1 = ApcomplexMath.pow(bottomlinefirstbr, this.s.divide(two));         Apcomplex D2 = (ApcomplexMath.exp(PI.multiply(t))).add(one);             Apcomplex denominator = D1.multiply(D2);         Apcomplex retval = numerator.divide(denominator);                //System.out.println("Integrand evaluated at "+t+ " is "+retval);         return retval;     }      /*      * Abel-Plana formulation integrand 25.5.11       */     private Apcomplex integrand2(Apfloat t)     {         Apcomplex numerator = ApcomplexMath.sin(this.s.multiply(ApcomplexMath.atan(t)));         Apcomplex bottomlinefirstbr = one.add(ApcomplexMath.pow(t, two));         Apcomplex D1 = ApcomplexMath.pow(bottomlinefirstbr, this.s.divide(two));         Apcomplex D2 = ApcomplexMath.exp(two.multiply(PI.multiply(t))).add(one.negate());             Apcomplex denominator = D1.multiply(D2);         Apcomplex retval = numerator.divide(denominator);                //System.out.println("Integrand evaluated at "+t+ " is "+retval);         return retval;     } } 

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A note on "correctness"

Note that in your answer - you are calling zeta(2+3i) and zeta(100) "correct" as compared to Wolfram when they exhibit errors of ~1e-10 and ~1e-9 respectively (they differ in the 10th and 9th decimal place), but you are worried about zeta(230+30i) because it exhibits an error of order 10e-14 in the imaginary component (38e-15 vs 5e-70 which are both very near zero). So in some senses the one you are calling "wrong" is closer to the Wolfram value than the ones you call "correct". Maybe you are worried that the leading digits are different, but this isn't really a measure of accuracy there.

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A final note

Unless you're doing it to learn about how functions behave and how floating point precision interacts with it - Don't do things this way. Even Apfloat's own documentation says:

This package is designed for extreme precision. The result might have a few digits less than you'd expect (about 10) and the last few (about 10) digits in ther result might be inaccurate. If you plan to use numbers with only a few hundred digits, use a program like PARI (it's free and available from ftp://megrez.math.u-bordeaux.fr) or a commercial program like Mathematica or Maple if possible.

I'd add mpmath in python to this list as a free alternative now.