See the previous and initial iteration.

I have incorporated almost all the suggestions by Peter Taylor:

• The actual method returns a BigInteger instead of String.
• The actual method can deal with Fibonacci numbers with negative indices, so there is no any need to throw IllegalArgumentException.
• The only base case that is checked is \$n = 0\$.
• fibonacciMatrix renamed to pow.
• main is now more sensible.

(The matrix power method is still recursive.)

See what I have:

import java.math.BigInteger; public class LargeFibonacciNumbers { public static BigInteger fibonacci(int n) { if (n == 0) { return BigInteger.ZERO; } BigInteger[][] matrix = new BigInteger[2][2]; matrix[0][0] = BigInteger.ONE; matrix[0][1] = BigInteger.ONE; matrix[1][0] = BigInteger.ONE; matrix[1][1] = BigInteger.ZERO; BigInteger tmp = pow(matrix, Math.abs(n))[0][1]; if (n > 0 || ((n & 1) == 1)) { return tmp; } else { return tmp.negate(); } } private static BigInteger[][] multiply(BigInteger[][] matrix1, BigInteger[][] matrix2) { BigInteger[][] ret = new BigInteger[2][2]; ret[0][0] = matrix1[0][0].multiply(matrix2[0][0]) .add(matrix1[0][1].multiply(matrix2[1][0])); ret[0][1] = matrix1[0][0].multiply(matrix2[0][1]) .add(matrix1[0][1].multiply(matrix2[1][1])); ret[1][0] = matrix1[1][0].multiply(matrix2[0][0]) .add(matrix1[1][1].multiply(matrix2[1][0])); ret[1][1] = matrix1[1][0].multiply(matrix2[0][1]) .add(matrix1[1][1].multiply(matrix2[1][1])); return ret; } private static BigInteger[][] pow(BigInteger[][] matrix, int n) { if (n == 1) { // End the recursion. return matrix; } BigInteger[][] tmp = pow(matrix, n >> 1); tmp = multiply(tmp, tmp); if ((n & 1) == 1) { tmp = multiply(matrix, tmp); } return tmp; } public static void main(String[] args) { if (args.length > 0) { int n; try { n = Integer.parseInt(args[0]); } catch (NumberFormatException ex) { System.err.println("Could not parse \"" + args[0] + "\" as an integer."); return; } System.out.println(fibonacci(n)); } else { System.out.println("Usage: java -jar File.jar N"); System.out.println("Where N is the index of the Fibonacci number " + "to compute."); } } } 

So, what do you think?