Converting double to BigDecimal in Java

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-15i

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) + 4f© + f(b));

double Q2 = h/12 * (f(a) + 4f(d) + 2f© + 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 + tt).pow(z.divide(TWO));

Complex D2 = new Complex(Math.pow(Math.E, 2.0Math.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).

#power-bi