Java- 寻找有关计算 min/max 函数或步长区间导数的建议

Question

寻求有关已变成 Java 噩梦的数学问题的建议。我扫描了网络，但找不到解决方案。我看过类似的程序，很遗憾找不到帮助。

问题总结：我希望在 Java 中实现一个方法，它将找到 Riemann-Siegel Z(t) 函数的最小值或最大值（我已经创建了用于计算的代码Z(t)) 或其导数的值。为了展示我正在尝试做的事情，从 0 < t < 100 开始的 Z(t) 图看起来像这样。

直接查看 Wolfram Alpha or in here 中的功能会使 "Java nightmare" 我看起来过于复杂。我描述的问题并不复杂，尽管这可能是由于我在数值分析方面缺乏经验。我想要做的事情的大纲是

在Java里面写一个方法，计算这个函数导数为零的所有地方（上图中，函数在0 < t之间大约有30个左右的值< 100).
在方法内部，定义步长间隔以通过下限和上限评估函数。
以下三种方法之一-用一种方法计算max/min，用两种方法计算max/min，或计算导数为零的值。
将这个添加到我现有的程序中（我做了一个测试程序，让问题更简单。测试程序看cos(x))

我浏览了互联网并找到了 this。我发现了很多其他不同的方法，但其中 none 似乎有效。提供的所有解决方案似乎只计算步长间隔内的一个 maximum/minimum/derivative。为了使用新方法，程序需要找到导数为零或函数具有最大值或最小值时的所有值。例如，cos(x) 在 0 < x < 50 之间有大约 16 个零（新方法会找到所有这些值）。

为了使这更容易，我创建了一个可以针对 cos(x) 函数进行分析的测试程序。

import java.math.*;

public class Test {
    public static void main(String[] args){
        Function cos = new Function () 
        {
        public double f(double x) {
        return Math.cos(x);
        }
    };


        //findRoots(cos, 1, 1000, 0.001); 
        findDerivative(cos, 1, 100, 0.001);
    }

    // Needed as a reference for the interpolation function.
    public static interface Function {
    public double f(double x);
    }

     private static int sign(double x) {
    if (x < 0.0)
            return -1;
        else if (x > 0.0)
            return 1;
        else
            return 0;
    }

     // Finds the roots of the specified function passed in with a lower bound,
    // upper bound, and step size.
    public static void findRoots(Function f, double lowerBound,
                  double upperBound, double step) {
    double x = lowerBound, next_x = x;
    double y = f.f(x), next_y = y;
    int s = sign(y), next_s = s;

    for (x = lowerBound; x <= upperBound ; x += step) {
        s = sign(y = f.f(x));
        if (s == 0) {
        System.out.println(x);
        } else if (s != next_s) {
        double dx = x - next_x;
        double dy = y - next_y;
        double cx = x - dx * (y / dy);
        System.out.println(cx);
        }
        next_x = x; next_y = y; next_s = s;
    }
    }

    public static void findDerivative(Function f, double lowerBound, 
            double upperBound, double step) {

    for (double x = lowerBound; x <= upperBound; x += step) {
        double fxstep = f.f(x);
        double fx = fxstep;
        fxstep = f.f(x+step);
        double dy = (fxstep - fx) / step;
        if (Math.abs(dy) < 0.001) {
            System.out.println("The x value is " + x + ". The value of the "
                    + "derivative is " + dy);
        }
    }

测试程序的目的是检查public static void findDerivative的方法是否正确。它有点工作，虽然它 returns 导数的近似值的两个值。 cos(x)的图形如下图

程序输出的值为

The x value is 3.140999999999764. The value of the derivative is -9.265358602572604E-5
The x value is 3.141999999999764. The value of the derivative is 9.073462475805982E-4
The x value is 6.282000000000432. The value of the derivative is 6.853070969592423E-4
The x value is 6.283000000000432. The value of the derivative is -3.1469280259432963E-4
The x value is 9.424000000000216. The value of the derivative is -2.7796075396935294E-4
The x value is 9.425000000000216. The value of the derivative is 7.220391380347024E-4
The x value is 12.564999999998475. The value of the derivative is 8.706142144987439E-4
The x value is 12.565999999998475. The value of the derivative is -1.2938563354047972E-4
The x value is 15.706999999996734. The value of the derivative is -4.632679163618647E-4
The x value is 15.707999999996733. The value of the derivative is 5.36731999623008E-4
The x value is 18.849000000000053. The value of the derivative is 5.592153640154862E-5
The x value is 18.850000000000055. The value of the derivative is -9.440782817726756E-4
The x value is 21.990000000003892. The value of the derivative is -6.485750521090239E-4
The x value is 21.991000000003893. The value of the derivative is 3.514248534397524E-4
The x value is 25.132000000007732. The value of the derivative is 2.4122869812792658E-4
The x value is 25.133000000007733. The value of the derivative is -7.587711848833223E-4
The x value is 28.27300000001157. The value of the derivative is -8.338821652076334E-4
The x value is 28.274000000011572. The value of the derivative is 1.6611769582119962E-4
The x value is 31.41500000001541. The value of the derivative is 4.2653585174967645E-4
The x value is 31.416000000015412. The value of the derivative is -5.734640621257725E-4
The x value is 34.55700000001016. The value of the derivative is -1.9189476674341677E-5
The x value is 34.55800000001016. The value of the derivative is 9.808103242914257E-4
The x value is 37.69800000000284. The value of the derivative is 6.118430110335638E-4
The x value is 37.69900000000284. The value of the derivative is -3.881568994001938E-4
The x value is 40.83999999999552. The value of the derivative is -2.0449666182642545E-4
The x value is 40.84099999999552. The value of the derivative is 7.955032111928162E-4
The x value is 43.9809999999882. The value of the derivative is 7.971501513326373E-4
The x value is 43.9819999999882. The value of the derivative is -2.028497212425151E-4
The x value is 47.12299999998088. The value of the derivative is -3.8980383987308187E-4
The x value is 47.123999999980875. The value of the derivative is 6.10196070671698E-4
The x value is 50.26399999997356. The value of the derivative is 9.824572642092022E-4
The x value is 50.264999999973554. The value of the derivative is -1.754253620145363E-5
The x value is 53.405999999966234. The value of the derivative is -5.75111004597062E-4
The x value is 53.40699999996623. The value of the derivative is 4.2488890927838696E-4
The x value is 56.54799999995891. The value of the derivative is 1.6776464961676396E-4
The x value is 56.54899999995891. The value of the derivative is -8.322352119671805E-4
The x value is 59.68899999995159. The value of the derivative is -7.604181495590723E-4
The x value is 59.68999999995159. The value of the derivative is 2.39581733230132E-4
The x value is 62.83099999994427. The value of the derivative is 3.530718295507995E-4
The x value is 62.831999999944266. The value of the derivative is -6.469280763310437E-4
The x value is 65.97199999995095. The value of the derivative is -9.457252543310091E-4
The x value is 65.97299999995096. The value of the derivative is 5.4274563066059045E-5
The x value is 69.11399999996596. The value of the derivative is 5.383789610791112E-4
The x value is 69.11499999996596. The value of the derivative is -4.616209549057615E-4
The x value is 72.25599999998096. The value of the derivative is -1.3103257845425986E-4
The x value is 72.25699999998096. The value of the derivative is 8.689672701400752E-4

虽然它需要计算两次导数，但由于 findDerivative 方法中的 Math.abs(dy) < 0.001，它变得很接近。以下解决方法均未成功。

求推荐，通过牛顿法求导。我不知道应用牛顿的任何方法，因为我不知道 Z(t) 的导数。
我在网上和其他网站上找到的所有程序都直接计算 "one" 区间内的最小值或最大值 [a, b]。在上面的图表和 Z(t) 函数的图表中，我正在寻找所有的最小值和最大值（或者，当函数为零时）。计算 [0, 100] 间隔之间的最小值或最大值没有帮助，我需要一种方法来计算所有这些值。
我本来低估了做这件事的难度

有人有什么建议吗？我该怎么做才能用 cos(x) 测试程序做到这一点？如果我得到这个工作，我可以自己去找出 Z(t) 程序。我花了很多时间思考这个问题并且失眠了。我自己一直想不出办法解决这个问题。

这是我用来计算一般值的 Z(t) 函数的方法（不需要理解下面的程序来解决这些困难）。

/**************************************************************************
**
**    Riemann-Siegel Formula for roots of Zeta(s) on critical line.
**
**************************************************************************
**    Axion004
**    07/31/2015
**
**    This program finds the roots of Zeta(s) using the well known Riemann-
**    Siegel formula. The Riemann–Siegel theta function is approximated 
**    using Stirling's approximation. It also uses an interpolation method to
**    locate zeroes. The coefficients for R(t) are handled by the Taylor
**    Series approximation originally listed by Haselgrove in 1960. It is 
**    necessary to use these coefficients in order to increase computational 
**    speed.
**************************************************************************/

public class SiegelMain{
    public static void main(String[] args){
        SiegelMain();
    }

    // Main method
    public static void SiegelMain() {
        Function RiemennSiegelZ = new Function () 
        {
        public double f(double x) {
        return RiemennZ(x, 4);
        }
    };
        System.out.println("Zeroes inside the critical line for " +
                "Zeta(1/2 + it). The t values are referenced below.");
        System.out.println();
        // Uncomment to find non-trivial zeroes for Zeta(1/2 + it)
    findRoots(RiemennSiegelZ, 1, 40000, 0.001);
        //findMax(RiemennSiegelZ, 1, 400, 0.001);
    }

    /**
     * Needed as a reference for the interpolation function.
    */
    public static interface Function {
    public double f(double x);
    }

    /**
     * The sign of a calculated double value.
     * @param x - the double value.
     * @return the sign in -1,  1, or 0 format.
    */
    private static int sign(double x) {
    if (x < 0.0)
            return -1;
        else if (x > 0.0)
            return 1;
        else
            return 0;
    }

    /**
     * Finds the roots of a specified function through interpolation.
     * @param f - the function
         * @param lowerBound - the lower bound of integration.
         * @param upperBound - the upper bound of integration.
         * @param step - the step for dx in [a:b]
     * @return the roots of the specified function.
    */
    public static void findRoots(Function f, double lowerBound,
                  double upperBound, double step) {
    double x = lowerBound, next_x = x;
    double y = f.f(x), next_y = y;
    int s = sign(y), next_s = s;

    for (x = lowerBound; x <= upperBound ; x += step) {
        s = sign(y = f.f(x));
        if (s == 0) {
        System.out.println(x);
        } else if (s != next_s) {
        double dx = x - next_x;
        double dy = y - next_y;
        double cx = x - dx * (y / dy);
        System.out.println(cx);
        }
        next_x = x; next_y = y; next_s = s;
    }
    }

    /**
     * Calculates the local maximum from a provided lower and upper bound.
     * @param f - the function
         * @param lowerBound - the lower bound of integration.
         * @param upperBound - the upper bound of integration.
         * @param step - the step for dx in [a:b]
     * @return the local maximum for the function.
    */
     public static void findMax(Function f, double lowerBound,
                  double upperBound, double step) {
    double x = lowerBound, next_x = x + step;
    double y = f.f(x), next_y = y + step;

    for (x = lowerBound; x <= upperBound ; x += step) {
            if (y > (next_y)) {
        System.out.println(y);
        }
        next_x = x; next_y = y;
    }
    }

    /**
     * Calculates the local minimum from a provided lower and upper bound.
     * @param f - the function
         * @param lowerBound - the lower bound of integration.
         * @param upperBound - the upper bound of integration.
         * @param step - the step for dx in [a:b]
     * @return the local minimum for the function.
    */
    public static double findMin(Function f, double lowerBound, double 
            upperBound, double step) {
    double minValue = f.f(lowerBound);

    for (double i=lowerBound; i <= upperBound; i+=step) {
        double currEval = f.f(i);
        if (currEval < minValue) {
            minValue = currEval;
        }
    }

        return minValue;
    }

    /**
     * Riemann-Siegel theta function using the approximation by the 
         * Stirling series.
     * @param t - the value of t inside the Z(t) function.
     * @return Stirling's approximation for theta(t).
    */
    public static double theta (double t) {
        return (t/2.0 * Math.log(t/(2.0*Math.PI)) - t/2.0 - Math.PI/8.0
                + 1.0/(48.0*Math.pow(t, 1)) + 7.0/(5760*Math.pow(t, 3)));
    }

    /**
     * Computes Math.Floor of the absolute value term passed in as t.
     * @param t - the value of t inside the Z(t) function.
     * @return Math.floor of the absolute value of t.
    */
    public static double fAbs(double t) {
        return Math.floor(Math.abs(t));

    }

    /**
     * Riemann-Siegel Z(t) function implemented per the Riemenn Siegel 
         * formula. See http://mathworld.wolfram.com/Riemann-SiegelFormula.html 
         * for details
     * @param t - the value of t inside the Z(t) function.
         * @param r - referenced for calculating the remainder terms by the
         * Taylor series approximations.
     * @return the approximate value of Z(t) through the Riemann-Siegel
         * formula
    */
    public static double RiemennZ(double t, int r) {

        double twopi = Math.PI * 2.0; 
        double val = Math.sqrt(t/twopi);
        double n = fAbs(val);
        double sum = 0.0;

        for (int i = 1; i <= n; i++) {
          sum += (Math.cos(theta(t) - t * Math.log(i))) / Math.sqrt(i);
        }
        sum = 2.0 * sum;

        double remainder;
        double frac = val - n; 
        int k = 0;
        double R = 0.0;

        // Necessary to individually calculate each remainder term by using
        // Taylor Series co-efficients. These coefficients are defined below.
        while (k <= r) {
            R = R + C(k, 2.0*frac-1.0) * Math.pow(t / twopi, 
                    ((double) k) * -0.5);
            k++;
        }

        remainder = Math.pow(-1, (int)n-1) * Math.pow(t / twopi, -0.25) * R;
        return sum + remainder;
    }

    /**
     * C terms for the Riemann-Siegel formula. See 
         * https://web.viu.ca/pughg/thesis.d/masters.thesis.pdf for details.
         * Calculates the Taylor Series coefficients for C0, C1, C2, C3, 
         * and C4. 
     * @param n - the number of coefficient terms to use.
         * @param z - referenced per the Taylor series calculations.
     * @return the Taylor series approximation of the remainder terms.
    */
    public static double C (int n, double z) {
        if (n==0) 
            return(.38268343236508977173 * Math.pow(z, 0.0) 
            +.43724046807752044936 * Math.pow(z, 2.0) 
            +.13237657548034352332 * Math.pow(z, 4.0) 
            -.01360502604767418865 * Math.pow(z, 6.0) 
            -.01356762197010358089 * Math.pow(z, 8.0) 
            -.00162372532314446528 * Math.pow(z,10.0) 
            +.00029705353733379691 * Math.pow(z,12.0) 
            +.00007943300879521470 * Math.pow(z,14.0) 
            +.00000046556124614505 * Math.pow(z,16.0) 
            -.00000143272516309551 * Math.pow(z,18.0) 
            -.00000010354847112313 * Math.pow(z,20.0) 
            +.00000001235792708386 * Math.pow(z,22.0) 
            +.00000000178810838580 * Math.pow(z,24.0) 
            -.00000000003391414390 * Math.pow(z,26.0) 
            -.00000000001632663390 * Math.pow(z,28.0) 
            -.00000000000037851093 * Math.pow(z,30.0) 
            +.00000000000009327423 * Math.pow(z,32.0) 
            +.00000000000000522184 * Math.pow(z,34.0) 
            -.00000000000000033507 * Math.pow(z,36.0) 
            -.00000000000000003412 * Math.pow(z,38.0)
            +.00000000000000000058 * Math.pow(z,40.0) 
            +.00000000000000000015 * Math.pow(z,42.0)); 
        else if (n==1) 
            return(-.02682510262837534703 * Math.pow(z, 1.0) 
            +.01378477342635185305 * Math.pow(z, 3.0) 
            +.03849125048223508223 * Math.pow(z, 5.0) 
            +.00987106629906207647 * Math.pow(z, 7.0) 
            -.00331075976085840433 * Math.pow(z, 9.0) 
            -.00146478085779541508 * Math.pow(z,11.0) 
            -.00001320794062487696 * Math.pow(z,13.0) 
            +.00005922748701847141 * Math.pow(z,15.0) 
            +.00000598024258537345 * Math.pow(z,17.0) 
            -.00000096413224561698 * Math.pow(z,19.0) 
            -.00000018334733722714 * Math.pow(z,21.0) 
            +.00000000446708756272 * Math.pow(z,23.0) 
            +.00000000270963508218 * Math.pow(z,25.0) 
            +.00000000007785288654 * Math.pow(z,27.0)
            -.00000000002343762601 * Math.pow(z,29.0) 
            -.00000000000158301728 * Math.pow(z,31.0) 
            +.00000000000012119942 * Math.pow(z,33.0) 
            +.00000000000001458378 * Math.pow(z,35.0) 
            -.00000000000000028786 * Math.pow(z,37.0) 
            -.00000000000000008663 * Math.pow(z,39.0) 
            -.00000000000000000084 * Math.pow(z,41.0) 
            +.00000000000000000036 * Math.pow(z,43.0) 
            +.00000000000000000001 * Math.pow(z,45.0)); 
      else if (n==2) 
            return(+.00518854283029316849 * Math.pow(z, 0.0) 
            +.00030946583880634746 * Math.pow(z, 2.0) 
            -.01133594107822937338 * Math.pow(z, 4.0) 
            +.00223304574195814477 * Math.pow(z, 6.0) 
            +.00519663740886233021 * Math.pow(z, 8.0) 
            +.00034399144076208337 * Math.pow(z,10.0) 
            -.00059106484274705828 * Math.pow(z,12.0) 
            -.00010229972547935857 * Math.pow(z,14.0) 
            +.00002088839221699276 * Math.pow(z,16.0) 
            +.00000592766549309654 * Math.pow(z,18.0) 
            -.00000016423838362436 * Math.pow(z,20.0) 
            -.00000015161199700941 * Math.pow(z,22.0) 
            -.00000000590780369821 * Math.pow(z,24.0) 
            +.00000000209115148595 * Math.pow(z,26.0) 
            +.00000000017815649583 * Math.pow(z,28.0) 
            -.00000000001616407246 * Math.pow(z,30.0) 
            -.00000000000238069625 * Math.pow(z,32.0) 
            +.00000000000005398265 * Math.pow(z,34.0) 
            +.00000000000001975014 * Math.pow(z,36.0) 
            +.00000000000000023333 * Math.pow(z,38.0) 
            -.00000000000000011188 * Math.pow(z,40.0) 
            -.00000000000000000416 * Math.pow(z,42.0) 
            +.00000000000000000044 * Math.pow(z,44.0) 
            +.00000000000000000003 * Math.pow(z,46.0)); 
      else if (n==3) 
            return(-.00133971609071945690 * Math.pow(z, 1.0) 
            +.00374421513637939370 * Math.pow(z, 3.0) 
            -.00133031789193214681 * Math.pow(z, 5.0) 
            -.00226546607654717871 * Math.pow(z, 7.0) 
            +.00095484999985067304 * Math.pow(z, 9.0) 
            +.00060100384589636039 * Math.pow(z,11.0) 
            -.00010128858286776622 * Math.pow(z,13.0) 
            -.00006865733449299826 * Math.pow(z,15.0) 
            +.00000059853667915386 * Math.pow(z,17.0) 
            +.00000333165985123995 * Math.pow(z,19.0)
            +.00000021919289102435 * Math.pow(z,21.0) 
            -.00000007890884245681 * Math.pow(z,23.0) 
            -.00000000941468508130 * Math.pow(z,25.0) 
            +.00000000095701162109 * Math.pow(z,27.0) 
            +.00000000018763137453 * Math.pow(z,29.0) 
            -.00000000000443783768 * Math.pow(z,31.0) 
            -.00000000000224267385 * Math.pow(z,33.0) 
            -.00000000000003627687 * Math.pow(z,35.0) 
            +.00000000000001763981 * Math.pow(z,37.0) 
            +.00000000000000079608 * Math.pow(z,39.0) 
            -.00000000000000009420 * Math.pow(z,41.0) 
            -.00000000000000000713 * Math.pow(z,43.0) 
            +.00000000000000000033 * Math.pow(z,45.0) 
            +.00000000000000000004 * Math.pow(z,47.0)); 
      else 
            return(+.00046483389361763382 * Math.pow(z, 0.0) 
            -.00100566073653404708 * Math.pow(z, 2.0) 
            +.00024044856573725793 * Math.pow(z, 4.0) 
            +.00102830861497023219 * Math.pow(z, 6.0) 
            -.00076578610717556442 * Math.pow(z, 8.0) 
            -.00020365286803084818 * Math.pow(z,10.0) 
            +.00023212290491068728 * Math.pow(z,12.0) 
            +.00003260214424386520 * Math.pow(z,14.0) 
            -.00002557906251794953 * Math.pow(z,16.0) 
            -.00000410746443891574 * Math.pow(z,18.0) 
            +.00000117811136403713 * Math.pow(z,20.0) 
            +.00000024456561422485 * Math.pow(z,22.0) 
            -.00000002391582476734 * Math.pow(z,24.0) 
            -.00000000750521420704 * Math.pow(z,26.0) 
            +.00000000013312279416 * Math.pow(z,28.0) 
            +.00000000013440626754 * Math.pow(z,30.0) 
            +.00000000000351377004 * Math.pow(z,32.0) 
            -.00000000000151915445 * Math.pow(z,34.0) 
            -.00000000000008915418 * Math.pow(z,36.0) 
            +.00000000000001119589 * Math.pow(z,38.0) 
            +.00000000000000105160 * Math.pow(z,40.0) 
            -.00000000000000005179 * Math.pow(z,42.0) 
            -.00000000000000000807 * Math.pow(z,44.0) 
            +.00000000000000000011 * Math.pow(z,46.0) 
            +.00000000000000000004 * Math.pow(z,48.0));
    }     
}

Answer 1

您似乎在尝试进行数值优化。 Apache Commons Math 库对这两个 optimization and root-finding 都有几种实现。即使您最终不得不编写自己的实现，在您自己实现它之前，首先使用库中提供的算法对您的解决方案进行原型设计以找到可行的算法可能会有所帮助。

Java- 寻找有关计算 min/max 函数或步长区间导数的建议

Java- Looking for advice on calculating the min/max for a function or the derivative in step intervals

java

max

min

numerical-methods

function-approximation