음이 아닌 실수 A 의 평방근 sqrt(A) 를 구하는 Heron 의 방법:

        반복함수  g(x) = (x + A/x) / 2   를 이용

 

실수 A 의 n제곱근 root(n, A) 를 구하는 Newton-Raphson 의 방법

        반복함수  g(x) = ((n-1)*x + A/(x**(n - 1))) / n    를 이용

n = 2 인 경우에는 Newton-Raphson 의 방법이 Heron 의 방법과 동일하다.

(참조. http://en.wikipedia.org/wiki/Newton's_method )

 

Java 언어에는 math.lang 패키지에 지수 계산 함수 Math.pow(double, double) 가 이미 구현되어 있다. 하지만 차후 필요한 데가 있을 것 같아서 이와 유사한 n 제곱 함수와 n 제곱근 함수를 구현해 보았다.

지수가 정수인 거듭제곱을 계산하는  함수도 nPow(), gPow, mPow() 세 개 구현해 놓았는데, 이들 세 함수는 절차적 언어의 성능상 재귀호출이 아니고 단순 반복 기법을 사용하는 함수이다. 이 세 함수 중 mPow() 의 성능이 가장 우수하다. 큰 지수의 경우 for 반복문의 반복회수를 따져 보면 성능 비교를 할 수 있을 것이다. (성능 비교를 위해 세 가지를 모두 소스에 남겨 두었다.) mPow() 함수는 n 제곱근을 구하는 재귀함수 newtonNthRoot(int, double) 의 구현에 사용되기도 한다. if ... else ... 구문이 많아 소스가 복잡하게 보일지 모르겠으나 이는 밑수나 지수가 음수이거나 0인 경우의 처리를 위함이다. 구현된 모든 함수의 구현에는 예외상황(예를 들어, 음수의 짝수 제곱근 같은 예외상황) 처리 과정이 있다.

아래의 소스는 대부분 버전의 JVM(자바가상기계) 위에서 컴파일 되고 실행되게 작성된 소스이다.

소스 첫 부분에

    final private static int MAX_ITER = 20000;
    final private static double M_EPSILON = 1.0e-15;

라고 선언하였으니 변수 MAX_ITER 와 M_EPSILON 는 상수는 아니지만 상수와 거의 같은 효과를 같는 클래스 소속 변수(스태틱 변수)이다. Java 언어에는 C 언어나 C++ 언어에서 말하는 상수 선언이 없다.

// Filename: TestNthRootApp.java
//
//            Approximate square roots, cubic roots and n-th roots of a given number.
//
// Compile: javac -d . TestNthRootApp.java
// Execute: java TestNthRootApp
//
// Date: 2013. 1. 6.
// Copyright (c) 2013 PH Kim  (pkim __AT__ scripts.pe.kr)


public class TestNthRootApp {

    final private static int MAX_ITER = 20000;
    final private static double M_EPSILON = 1.0e-15;

    /**
     * Compute the n-th root of x to a given scale, x > 0.
     */
    public static double nPow(double a, int n) {
        if (n > 0) {
            if (n == 1)
                return a;
            else {
                if (a == 0.0 || a == 1.0) {
                    return a;
                }
                else if (a == -1.0) {
                    if (n % 2 == 1)
                        return -1.0;
                    else
                        return 1.0;
                }
                else if (a < 0.0) {
                    if (n % 2 == 1)
                        return -nPow(-a, n);
                    else
                        return nPow(-a, n);
                }
                else {
                    double y = 1.0;
                    for (int i = 0; i < n; i++) {
                        y *= a;
                    }
                    return y;
                }
            }
        }
        else if (n == 0) {
            return 1.0;
        }
        else {      //  when n < 0
            if (a == 0.0)
                throw new RuntimeException("Negative powering exception of zero.");
            else {
                if (n == -1)
                    return 1.0/a;
                else
                    return 1.0/nPow(a, -n);
            }
        }
    }

 

    /**
     * Compute the n-th root of x to a given scale, x > 0.
     */
    public static double gPow(double a, int n) {
        if (n > 0) {
            if (n == 1)
                return a;
            else {
                if (a == 0.0 || a == 1.0) {
                    return a;
                }
                else if (a == -1.0) {
                    if (n % 2 == 1)
                        return -1.0;
                    else
                        return 1.0;
                }
                else if (a < 0.0) {
                    if (n % 2 == 1)
                        return -gPow(-a, n);
                    else
                        return gPow(-a, n);
                }
                else {

                    double y = 1.0;
                    double r = a;
                    int m = 8*4 - 1;            ///  8*sizeof(int) - 1;
                    int one = 1;
                    for (int i = 0; i < m; i++) {
                        if ((n & one) == 0) {
                            y *= 1.0;
                        }
                        else {
                            y *= r;
                        }
                        r = r*r;
                        one <<= 1;
                        if (one > n)
                            break;
                    }
                    return y;
                }
            }
        }
        else if (n == 0) {
            return 1.0;
        }
        else {      //  when n < 0
            if (a == 0.0)
                throw new RuntimeException("Negative powering exception of zero.");
            else {
                if (n == -1)
                    return 1.0/a;
                else
                    return 1.0/gPow(a, -n);
            }
        }
    }


    /**
     * Compute the n-th root of x to a given scale, x > 0.
     */
    public static double mPow(double a, int n) {
        if (n > 0) {
            if (n == 1)
                return a;
            else {
                if (a == 0.0 || a == 1.0) {
                    return a;
                }
                else if (a == -1.0) {
                    if (n % 2 == 1)
                        return -1.0;
                    else
                        return 1.0;
                }
                else if (a < 0.0) {
                    if (n % 2 == 1)
                        return -mPow(-a, n);
                    else
                        return mPow(-a, n);
                }
                else {

                    double y = 1.0;
                    double r = a;
                    int m = n;
                    while (m > 0) {
                        if ((m & 0x1) == 1) {
                            y *= r;
                        }
                        r = r*r;
                        m >>= 1;
                    }
                    return y;
                }
            }
        }
        else if (n == 0) {
            return 1.0;
        }
        else {      //  when n < 0
            if (a == 0.0)
                throw new RuntimeException("Negative powering exception of zero.");
            else {
                if (n == -1)
                    return 1.0/a;
                else
                    return 1.0/mPow(a, -n);
            }
        }
    }

 

    /**
     * Compute the square root of x to a given scale, x > 0.
     */
    public static double heronSqrt(double a) {
        if (a < 0.0) {
            throw new RuntimeException("Cannot find the sqrt of a negative number.");
        }
        else if (a == 0.0 || a == 1.0) {
            return a;
        }
        else {
            double x1 = a;
            double x2 = (x1 + a/x1)/2.0;
            double er = x1 - x2;
            int counter = 0;
            while (x1 + er != x1) {
                x1 = x2;
                x2 = (x1 + a/x1)/2.0;
                er = x1 - x2;
                if (Math.abs(er) < Math.abs(M_EPSILON*x1))
                    break;
                counter++;
                if (counter > MAX_ITER)
                    break;
            }
            if (counter >= MAX_ITER)
                throw new RuntimeException("Inaccurate sqrt exception by too many iterations.");
            return x2;
        }
    }

    /**
     * Compute the cubic root of x to a given scale, x > 0.
     */
    public static double newtonCbrt(double a) {
        if (a == 0.0 || a == 1.0 || a == -1.0) {
            return a;
        }
        else if (a < 0.0) {
            return -newtonCbrt(-a);
        }
        else {
            double x1 = a;
            double x2 = (2.0*x1 + a/(x1*x1))/3.0;
            double er = x1 - x2;
            int counter = 0;
            while (x1 + er != x1) {
                x1 = x2;
                x2 = (2.0*x1 + a/(x1*x1))/3.0;
                er = x1 - x2;
                if (Math.abs(er) < Math.abs(M_EPSILON*x1))
                    break;
                counter++;
                if (counter > MAX_ITER)
                    break;
            }
            if (counter >= MAX_ITER)
                throw new RuntimeException("Inaccurate cbrt exception by too many iterations.");
            return x2;
        }
    }

    /**
     * Compute the n-th root of x to a given scale, x > 0.
     */
    public static double newtonNthRoot(int n, double a) {
        if (n == 0) {
            return 1.0;
        }
        else if (n == 1) {
            return a;
        }
        else if (n > 0) {
            if (a == 0.0 || a == 1.0) {
                return a;
            }
            else if (a == -1.0) {
                if (n % 2 == 1)
                    return a;
                else
                    throw new RuntimeException("Cannot find the even n-th root of a negative number.");
            }
            else if (a < 0.0) {
                if (n % 2 == 1)
                    return -newtonNthRoot(n, -a);
                else
                    throw new RuntimeException("Cannot find the even n-th root of a negative number.");
            }
            else if (a < 1.0) {
                return 1.0/newtonNthRoot(n, 1.0/a);
            }
            else {
                double x1 = a;
                double xn = mPow(x1, n - 1);
                double x2 = ((n - 1)*x1 + a/xn)/n;
                double er = x1 - x2;
                int counter = 0;
                while (x1 + er != x1) {
                    x1 = x2;
                    xn = mPow(x1, n - 1);
                    x2 = ((n - 1)*x1 + a/xn)/n;
                    er = x1 - x2;
                    if (Math.abs(er) < Math.abs(M_EPSILON*x1))
                        break;
                    counter++;
                    if (counter > MAX_ITER)
                        break;
                }
                if (counter >= MAX_ITER)
                    throw new RuntimeException("Inaccurate n-th root exception by too many iterations.");
                return x2;
            }
        }
        else {
            if (a == 0.0) {
                throw new RuntimeException("Cannot find the negative n-th root of zero.");
            }
            else {
                return 1.0/newtonNthRoot(-n, a);
            }
        }
    }


    public static void main(String[] args) {

        double x = 16.0;
        double u = Math.sqrt(x);

        System.out.println("[ Testing heronSqrt(double) ]--------------------");
        System.out.println("x = " + x );
        System.out.println("u = sqrt(" + x + ") = " + u );
        double y = heronSqrt(x);
        System.out.println("y = heronSqrt(" + x + ") = " + y );
        System.out.println("y*y = " + y*y );
        System.out.println();

        System.out.println("[ Testing newtonCbrt(double) ]--------------------" );
        x = -216.0;
        System.out.println("x = " + x );
        System.out.println("-exp(log(-x)/3.0) = " + -Math.exp(Math.log(-x)/3.0) );
        double w = newtonCbrt(x);
        System.out.println("w = newtonCbrt(" + x + ") = " + w );
        System.out.println("w*w*w = " + w*w*w );
        System.out.println();

        x = 729000000000.0;
        System.out.println("x = " + x );
        System.out.println("exp(log(x)/3.0) = " + Math.exp(Math.log(x)/3.0) );
        w = newtonCbrt(x);
        System.out.println("w = newtonCbrt(" + x + ") = " + w );
        System.out.println("w*w*w = " + w*w*w );
        System.out.println();

        System.out.println("[ Testing newtonNthRoot(int, double) ]--------------------" );
        double z = newtonNthRoot(3, x);
        System.out.println("x = " + x );
        System.out.println("z = newtonNthRoot(3, " + x + ") = " + z );
        System.out.println("z*z*z = " + z*z*z );
        System.out.println();

        x = 12960000000000000000.0;
        z = newtonNthRoot(4, x);
        System.out.println("x = " + x );
        System.out.println("z = newtonNthRoot(4, x) = newtonNthRoot(4, " + x + ") = " + z );
        System.out.println("z*z*z*z = " + z*z*z*z );
        System.out.println();

        x = 1.0/12960000000000000000.0;
        z = newtonNthRoot(4, x);
        System.out.println("x = " + x );
        System.out.println("exp(log(x)/4.0) = " + Math.exp(Math.log(x)/4.0) );
        System.out.println("z = newtonNthRoot(4, x) = newtonNthRoot(4, " + x + ") = " + z );
        System.out.println("z*z*z*z = " + z*z*z*z );
        System.out.println();


        try {
            x = -4.0;
            System.out.println("[ Test Exception heronSqrt(double) ]--------------------" );
            System.out.println("x = " + x );
            System.out.println("Calculating heronSqrt(" + x + ")" );
            y = heronSqrt(x);
            System.out.println("y = heronSqrt(" + x + ") = " + y );
            System.out.println("y*y = " + y*y );
            System.out.println();
        }
        catch (Exception ex) {
            System.out.println(ex.getMessage() + "\n" + "Caught some exception in calculating heronSqrt(" + x + ")" );
            System.out.println();
        }


        try {
            x = -4.0;
            System.out.println("[ Test Exception in newtonCbrt(double) ]--------------------" );
            System.out.println("x = " + x );
            System.out.println("Calculating newtonCbrt(" + x + ")" );
             = newtonCbrt(x);
            System.out.println("y = newtonCbrt(" + x + ") = " + y );
            System.out.println("y*y*y = " + y*y*y );
            System.out.println();
        }
        catch (Exception ex) {
            System.out.println(ex.getMessage() + "\n" + "Caught some exception in calculating newtonCbrt(" + x + ")");
            System.out.println();
        }


        System.out.println("[ Test calculations by powering ]-----------------------------" );
        x = 200.0;
        z = newtonNthRoot(10, x);
        System.out.println("x = " + x );
        System.out.println("exp(log(x)/10.0) = " + Math.exp(Math.log(x)/10.0) );
        System.out.println("z = newtonNthRoot(10, x) = newtonNthRoot(10, " + x + ") = " + z );
        System.out.println("pow(z, 10) = " + Math.pow(z, 10) );
        System.out.println();

        x = 3001.0;
        z = newtonNthRoot(99, x);
        System.out.println("x = " + x );
        System.out.println("exp(log(x)/99.0) = " + Math.exp(Math.log(x)/99.0) );
        System.out.println("z = newtonNthRoot(99, x) = newtonNthRoot(99, " + x + ") = " + z );
        System.out.println("pow(z, 99) = " + Math.pow(z, 99) );
        System.out.println();

        x = 3001.0;
        z = newtonNthRoot(-99, x);
        System.out.println("x = " + x );
        System.out.println("exp(log(x)/-99.0) = " + Math.exp(Math.log(x)/-99.0) );
        System.out.println("z = newtonNthRoot(-99, x) = newtonNthRoot(-99, " + x + ") = " + z );
        System.out.println("1.0/pow(z, 99) = " + 1.0/Math.pow(z, 99) );
        System.out.println();


        System.out.println("2.1**2.1 = pow(2.1, 2.1) = "  + Math.pow(2.1, 2.1) );
        System.out.println("2.1**(-2.1) = pow(2.1, -2.1) = "  + Math.pow(2.1, -2.1) );
        System.out.println("2.1**2.1 * 2.1**(-2.1) = pow(2.1, 2.1) * pow(2.1, -2.1) = "  + Math.pow(2.1, 2.1)*Math.pow(2.1, -2.1) );
        System.out.println("2.1**2.1 = exp(2.1*log(2.1)) = "  + Math.exp(2.1*Math.log(2.1)) );
        System.out.println("2.1**(-2.1) = exp(-2.1*log(2.1)) = " + Math.exp(-2.1*Math.log(2.1)) );
        System.out.println("2.1**2.1 * 2.1**(-2.1) = exp(2.1*log(2.1)) * exp(-2.1*log(2.1)) = "  + Math.exp(2.1*Math.log(2.1)) * Math.exp(-2.1*Math.log(2.1)) );
        System.out.println();


        int k = 301;
        x = -1.029;
        double t1 = nPow(x, k);
        double t2 = gPow(x, k);
        double t3 = mPow(x, k);
        System.out.println("t1 = nPow(" + x + ", " + k + ") = " + t1 );
        System.out.println("t2 = gPow(" + x + ", " + k + ") = " + t2 );
        System.out.println("t3 = mPow(" + x + ", " + k + ") = " + t3 );
        System.out.println("t1 / t2 = " + (t1 / t2) );
        System.out.println("t1 - t2 = " + (t1 - t2) );
        System.out.println("t1 == t2 ? " + ((t1 == t2) ? "yes" : "no") );
        System.out.println("t1 / t3 = " + (t1 / t3) );
        System.out.println("t1 - t3 = " + (t1 - t3) );
        System.out.println("t1 == t3 ? " + ((t1 == t3) ? "yes" : "no") );
        System.out.println("t2 / t3 = " + (t2 / t3) );
        System.out.println("t2 - t3 = " + (t2 - t3) );
        System.out.println("t2 == t3 ? " + ((t2 == t3) ? "yes" : "no") );
        System.out.println();

        System.out.println("Done.");
    }
}

/*
[ Testing heronSqrt(double) ]--------------------
x = 16.0
u = sqrt(16.0) = 4.0
y = heronSqrt(16.0) = 4.0
y*y = 16.0

[ Testing newtonCbrt(double) ]--------------------
x = -216.0
-exp(log(-x)/3.0) = -6.000000000000001
w = newtonCbrt(-216.0) = -6.0
w*w*w = -216.0

x = 7.29E11
exp(log(x)/3.0) = 9000.000000000004
w = newtonCbrt(7.29E11) = 9000.0
w*w*w = 7.29E11

[ Testing newtonNthRoot(int, double) ]--------------------
x = 7.29E11
z = newtonNthRoot(3, 7.29E11) = 9000.0
z*z*z = 7.29E11

x = 1.296E19
z = newtonNthRoot(4, x) = newtonNthRoot(4, 1.296E19) = 60000.0
z*z*z*z = 1.296E19

x = 7.716049382716049E-20
exp(log(x)/4.0) = 1.666666666666666E-5
z = newtonNthRoot(4, x) = newtonNthRoot(4, 7.716049382716049E-20) = 1.6666666666
666667E-5
z*z*z*z = 7.716049382716051E-20

[ Test Exception heronSqrt(double) ]--------------------
x = -4.0
Calculating heronSqrt(-4.0)
Cannot find the sqrt of a negative number.
Caught some exception in calculating heronSqrt(-4.0)

[ Test Exception in newtonCbrt(double) ]--------------------
x = -4.0
Calculating newtonCbrt(-4.0)
y = newtonCbrt(-4.0) = -1.5874010519681994
y*y*y = -3.999999999999999

[ Test calculations by powering ]-----------------------------
x = 200.0
exp(log(x)/10.0) = 1.6986464646342472
z = newtonNthRoot(10, x) = newtonNthRoot(10, 200.0) = 1.6986464646342472
pow(z, 10) = 199.9999999999999

x = 3001.0
exp(log(x)/99.0) = 1.0842361893258805
z = newtonNthRoot(99, x) = newtonNthRoot(99, 3001.0) = 1.0842361893258805
pow(z, 99) = 3000.9999999999955

x = 3001.0
exp(log(x)/-99.0) = 0.9223082662659932
z = newtonNthRoot(-99, x) = newtonNthRoot(-99, 3001.0) = 0.9223082662659932
1.0/pow(z, 99) = 3001.000000000004

2.1**2.1 = pow(2.1, 2.1) = 4.749638091742242
2.1**(-2.1) = pow(2.1, -2.1) = 0.21054235726688475
2.1**2.1 * 2.1**(-2.1) = pow(2.1, 2.1) * pow(2.1, -2.1) = 0.9999999999999999
2.1**2.1 = exp(2.1*log(2.1)) = 4.749638091742242
2.1**(-2.1) = exp(-2.1*log(2.1)) = 0.21054235726688478
2.1**2.1 * 2.1**(-2.1) = exp(2.1*log(2.1)) * exp(-2.1*log(2.1)) = 1.0

t1 = nPow(-1.029, 301) = -5457.92801577163
t2 = gPow(-1.029, 301) = -5457.928015771692
t3 = mPow(-1.029, 301) = -5457.928015771692
t1 / t2 = 0.9999999999999887
t1 - t2 = 6.184563972055912E-11
t1 == t2 ? no
t1 / t3 = 0.9999999999999887
t1 - t3 = 6.184563972055912E-11
t1 == t3 ? no
t2 / t3 = 1.0
t2 - t3 = 0.0
t2 == t3 ? yes

Done.
*/


 

 

Posted by Scripter

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  1. oroblast 2014.06.01 14:06  댓글주소  수정/삭제  댓글쓰기

    담아갑니다.
    좋은 내용 감사~^^~
    http://blog.naver.com/oroblast/220017051412