Java-寻找关于计算函数的最小值/最大值或步长间隔中的导数的建议
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内容图文
寻找有关数学问题的建议,这已成为Java的噩梦.我扫描了网络,找不到解决方案.我看过类似的程序,遗憾的是找不到帮助.
问题摘要：我希望在Java中实现一个方法,它可以找到Riemann-Siegel Z(t)函数的最小值或最大值(我已经创建了计算Z(t)的代码)或者值它的衍生物.为了显示我想要做的事情,Z(t)的图表从0 <0. t< 100看起来像这样.
直接回顾Wolfram Alpha或here中的功能使得“Java噩梦”让我看起来过于复杂.我所描述的问题并不是非常复杂,尽管可能是由于我在数值分析方面缺乏经验.我要做的一般大纲是
>在Java中编写一个方法来计算该函数的导数为零的所有位置(在上图中,该函数在0 <t <100之间具有大约30个左右的值).<br="">>在方法内部,定义步骤间隔以通过下限和上限评估函数.
>以下三种方法之一 – 在一种方法中计算最大值/最小值,在两种方法中计算最大值/最小值,或计算导数为零的值.
>将其添加到我现有的程序中(我已经制作了一个测试程序,以使问题更容易.测试程序查看cos(x))
我扫描了互联网,发现了this.我发现了很多其他不同的方法,但这些方法似乎都没有用.所提供的所有解决方案似乎仅计算步进间隔内的一个最大值/最小值/导数.为了使用新方法,程序需要找到导数为零的所有值,或者当函数具有最大值或最小值时.作为示例,cos(x)在0 <0之间具有大约16个零. x< 50(新方法将找到所有这些值). 为了简化这一过程,我创建了一个可以根据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.虽然它为导数的近似值返回两个值,但它有点有用. cos(x)的图表如下所示.
程序输出的值是
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
它接近,虽然它需要计算导数两次,因为Math.abs(dy)<在findDerivative方法中有0.001.以下方法都是不成功的.
>有人建议通过牛顿法计算导数.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直接在[a,b]的间隔内仅计算“一”最小值或最大值.在上图和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).计算间隔[0,100]之间的一个最小值或最大值没有帮助,我需要有一个方法来计算所有这些.
>我原本低估了这样做的难度.
有没有人有建议？我可以用cos(x)测试程序做到这一点吗？如果我有这个工作,我可以自己去找出Z(t)程序.我花了很多时间思考这个并且失去了一些睡眠.我自己无法想办法解决这个问题.
这是我用来计算一般值的Z(t)函数(没有必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));
}
}
解决方法:
看起来你正在尝试进行数值优化. Apache Commons Math库有optimization和root-finding的几个实现.即使您最终必须编写自己的实现,使用库中可用的算法首先对您的解决方案进行原型设计可能会有所帮助,以便在实现它之前找到一个可用的算法.你自己.
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