NSDecimalNumber+SquareRoot.h

//
//  NSDecimalNumber+SquareRoot.h
//  CubeRoot
//
//  Created by Michael Taylor on 10-06-21.
//

#import <Cocoa/Cocoa.h>

@interface NSDecimalNumber (SquareRoot)

- (NSDecimalNumber *) squareRoot;
- (NSDecimalNumber *) cubeRoot;
- (NSDecimalNumber *) abs;
- (NSDecimalNumber *) sign;
- (BOOL) isEquivalentTo:(NSDecimalNumber*)number;

@end

NSDecimalNumber+SquareRoot.m

//
//  NSDecimalNumber+SquareRoot.m
//  CubeRoot
//
//  Created by Michael Taylor on 10-06-21.
//

#import "NSDecimalNumber+SquareRoot.h"

static NSString * Tolerance = @"0.0000000000000001";

static NSDecimalNumber * MinusOne = nil;

@implementation NSDecimalNumber (SquareRoot)

- (NSDecimalNumber *) squareRoot
{
    NSComparisonResult comparison = [self compare:[NSDecimalNumber zero]];
    if (comparison == NSOrderedAscending)
        return [NSDecimalNumber notANumber];
    else if (comparison == NSOrderedSame)
	return [NSDecimalNumber zero];
    
    NSDecimalNumber *half =
    [[NSDecimalNumber one] decimalNumberByDividingBy:[NSDecimalNumber decimalNumberWithString:@"2"]];
    NSDecimalNumber *guess =
    [[self decimalNumberByAdding:[NSDecimalNumber one]]
     decimalNumberByMultiplyingBy:half];
    NSDecimalNumber * next_guess;
    
    @try
    {
        while (TRUE)
        {
	    next_guess = [[[self decimalNumberByDividingBy:guess]
			   decimalNumberByAdding:guess]
			  decimalNumberByMultiplyingBy:half];
	    
	    if ([next_guess isEquivalentTo:guess])
		// We are close enough
		break;
	    
	    guess = next_guess;
        }
    }
    @catch (NSException *exception)
    {
        // deliberately ignore exception and assume the last guess is good enough
    }
    
    NSDecimalNumber * int_value = (NSDecimalNumber*)[NSDecimalNumber numberWithInt:[guess intValue]];
    if ([guess isEquivalentTo:int_value])
	return int_value;
    else
	return guess;
}

- (NSDecimalNumber *) cubeRoot
{
    NSComparisonResult comparison = [self compare:[NSDecimalNumber zero]];
    if (comparison == NSOrderedSame)
	return [NSDecimalNumber zero];
    
    BOOL negative = NO;
    NSDecimalNumber * value = [self abs];
    NSDecimalNumber * guess, * next_guess, * two, * three;
    
    if ([self compare:[NSDecimalNumber zero]] == NSOrderedAscending)
    {
        negative = YES;
	value = [self abs];
    }
    
    guess = [self sign];
    two = [NSDecimalNumber decimalNumberWithString:@"2"];
    three = [NSDecimalNumber decimalNumberWithString:@"3"];
    
    @try
    {
        while (TRUE)
        {
	    // (x/y^2 + 2y)/3
	    next_guess = [[[self decimalNumberByDividingBy:
			    [guess decimalNumberByMultiplyingBy:guess]] 
			   decimalNumberByAdding:
			   [two decimalNumberByMultiplyingBy:guess]] 
			  decimalNumberByDividingBy:three];
	    
	    if ([next_guess isEquivalentTo:guess])
		// We are close enough
		break;
		
	    guess = next_guess;
        }
    }
    @catch (NSException *exception)
    {
        // deliberately ignore exception and assume the last guess is good enough
    }
    
    NSDecimalNumber * int_value = (NSDecimalNumber*)[NSDecimalNumber numberWithInt:[guess intValue]];
    if ([guess isEquivalentTo:int_value])
	return int_value;
    else
	return guess;
}

// Absolute value
- (NSDecimalNumber *) abs
{
    if (!MinusOne)
	MinusOne = [NSDecimalNumber decimalNumberWithString:@"-1"];
    
    if ([self compare:[NSDecimalNumber zero]] == NSOrderedAscending)
	return [self decimalNumberByMultiplyingBy:MinusOne];
    else
	return self;
}
		 
// Sign returns either 1.0 or -1.0 depending on the sign of the value
- (NSDecimalNumber *) sign
{
    if (!MinusOne)
	MinusOne = [NSDecimalNumber decimalNumberWithString:@"-1"];
    
    if ([self compare:[NSDecimalNumber zero]] == NSOrderedAscending)
	return MinusOne;
    else
	return [NSDecimalNumber one];
}

// Equivalence test
- (BOOL) isEquivalentTo:(NSDecimalNumber*)number
{
    static NSDecimalNumber * tolerance = nil;
    
    if (!tolerance)
	tolerance = [NSDecimalNumber decimalNumberWithString:Tolerance];
    
    NSDecimalNumber * difference = [[self decimalNumberBySubtracting:number] abs];    
    return [difference compare:tolerance] == NSOrderedAscending;
}

@end

A simple category that extends NSDecimalNumber to support square and cuberoots using Netwon's method. It is loosely based on some code from Matt Gallagher.

The advantages of this version over Matt's is that, instead of a fixed number if iterations, it will iterate until within a certain tolerance which you can set. Also, cube roots are supported. And, the code is presented as Category to make it easier to work with. I threw in a couple of other useful methods as a bonus.