Logarithm for BigInteger
I have a BigInteger number, for example beyond 264.
Now i want to calculate the logarithm of that BigInteger number, but the method BigInteger.log() does not exist. How do I calculate the (natural) logarithm of my large BigInteger value?
If you want to support arbitrarily big integers, it's not safe to just do
Math.log(bigInteger.doubleValue());
because this would fail if the argument exceeds the double range (about 2^1024 or 10^308, i.e. more than 300 decimal digits ).
Here's my own class that provides the methods
double logBigInteger(BigInteger val);
double logBigDecimal(BigDecimal val);
BigDecimal expBig(double exponent);
BigDecimal powBig(double a, double b);
They work safely even when the BigDecimal/BigInteger are too big (or too small) to be representable as a double type.
import java.math.*;
/**
* Provides some mathematical operations on {@code BigDecimal} and {@code BigInteger}.
* Static methods.
*/
public class BigMath {
public static final double LOG_2 = Math.log(2.0);
public static final double LOG_10 = Math.log(10.0);
// numbers greater than 10^MAX_DIGITS_10 or e^MAX_DIGITS_E are considered unsafe ('too big') for floating point operations
private static final int MAX_DIGITS_10 = 294;
private static final int MAX_DIGITS_2 = 977; // ~ MAX_DIGITS_10 * LN(10)/LN(2)
private static final int MAX_DIGITS_E = 677; // ~ MAX_DIGITS_10 * LN(10)
/**
* Computes the natural logarithm of a {@link BigInteger}
* <p>
* Works for really big integers (practically unlimited), even when the argument
* falls outside the {@code double} range
* <p>
*
*
* @param val Argument
* @return Natural logarithm, as in {@link java.lang.Math#log(double)}<br>
* {@code Nan} if argument is negative, {@code NEGATIVE_INFINITY} if zero.
*/
public static double logBigInteger(BigInteger val) {
if (val.signum() < 1)
return val.signum() < 0 ? Double.NaN : Double.NEGATIVE_INFINITY;
int blex = val.bitLength() - MAX_DIGITS_2; // any value in 60..1023 works here
if (blex > 0)
val = val.shiftRight(blex);
double res = Math.log(val.doubleValue());
return blex > 0 ? res + blex * LOG_2 : res;
}
/**
* Computes the natural logarithm of a {@link BigDecimal}
* <p>
* Works for really big (or really small) arguments, even outside the double range.
*
* @param val Argument
* @return Natural logarithm, as in {@link java.lang.Math#log(double)}<br>
* {@code Nan} if argument is negative, {@code NEGATIVE_INFINITY} if zero.
*/
public static double logBigDecimal(BigDecimal val) {
if (val.signum() < 1)
return val.signum() < 0 ? Double.NaN : Double.NEGATIVE_INFINITY;
int digits = val.precision() - val.scale();
if (digits < MAX_DIGITS_10 && digits > -MAX_DIGITS_10)
return Math.log(val.doubleValue());
else
return logBigInteger(val.unscaledValue()) - val.scale() * LOG_10;
}
/**
* Computes the exponential function, returning a {@link BigDecimal} (precision ~ 16).
* <p>
* Works for very big and very small exponents, even when the result
* falls outside the double range.
*
* @param exponent Any finite value (infinite or {@code Nan} throws {@code IllegalArgumentException})
* @return The value of {@code e} (base of the natural logarithms) raised to the given exponent,
* as in {@link java.lang.Math#exp(double)}
*/
public static BigDecimal expBig(double exponent) {
if (!Double.isFinite(exponent))
throw new IllegalArgumentException("Infinite not accepted: " + exponent);
// e^b = e^(b2+c) = e^b2 2^t with e^c = 2^t
double bc = MAX_DIGITS_E;
if (exponent < bc && exponent > -bc)
return new BigDecimal(Math.exp(exponent), MathContext.DECIMAL64);
boolean neg = false;
if (exponent < 0) {
neg = true;
exponent = -exponent;
}
double b2 = bc;
double c = exponent - bc;
int t = (int) Math.ceil(c / LOG_10);
c = t * LOG_10;
b2 = exponent - c;
if (neg) {
b2 = -b2;
t = -t;
}
return new BigDecimal(Math.exp(b2), MathContext.DECIMAL64).movePointRight(t);
}
/**
* Same as {@link java.lang.Math#pow(double,double)} but returns a {@link BigDecimal} (precision ~ 16).
* <p>
* Works even for outputs that fall outside the {@code double} range.
* <br>
* The only limitation is that {@code b * log(a)} cannot exceed the {@code double} range.
*
* @param a Base. Should be non-negative
* @param b Exponent. Should be finite (and non-negative if base is zero)
* @return Returns the value of the first argument raised to the power of the second argument.
*/
public static BigDecimal powBig(double a, double b) {
if (!(Double.isFinite(a) && Double.isFinite(b)))
throw new IllegalArgumentException(
Double.isFinite(b) ? "base not finite: a=" + a : "exponent not finite: b=" + b);
if (b == 0)
return BigDecimal.ONE;
else if (b == 1)
return BigDecimal.valueOf(a);
if (a <= 0) {
if (a == 0) {
if (b >= 0)
return BigDecimal.ZERO;
else
throw new IllegalArgumentException("0**negative = infinite b=" + b);
} else
throw new IllegalArgumentException("negative base a=" + a);
}
double x = b * Math.log(a);
if (Math.abs(x) < MAX_DIGITS_E)
return BigDecimal.valueOf(Math.pow(a, b));
else
return expBig(x);
}
}
I had some help from google but apparently you don't need to apply log to your very big BigInteger numbers directly, since it can be broken down in the following way:
928 = 1000 * 0.928
lg 928 = lg 1000 + lg 0.928 = 3 + lg 0.928
Your problem is therefore reduced to the computation/approximation of logarithms that allow for arbitrary increasing precision, maybe math.stackexchange.com?
Convert it into a BigDecimal liek this:
new BigDecimal(val); // where val is a BigInteger
and call log from BigDecimalUtils on it :D
How accurate do you need it to be? If you only need 15 digits of accuracy you can do
BigInteger bi =
double log = Math.log(bi.doubleValue());
This would work for values up to 1023 bits. After that the value would not fit into a double anymore.
If you can use Google Guava, and only require base 2 or base 10 log, you can use methods from Guava's BigIntegerMath class.
If you need a different base, you can always use the logarithm change-of-base formula to convert from one of these, to the one you need.