[从头学数学] 第169节 函数的应用

剧情提要：
[机器小伟]在[工程师阿伟]的陪同下进入了结丹初期的修炼，
这次要修炼的目标是[函数的应用]。

正剧开始：

星历2016年04月09日 11:38:05, 银河系厄尔斯星球中华帝国江南行省。
[工程师阿伟]正在和[机器小伟]一起研究[函数的应用]。

<span style="font-size:18px;">function funTask(x) {
	return Math.log(x)/Math.log(Math.E)+2*x-6;
}


		var r = 20;      
        config.setSector(1,1,1,1);        
        config.graphPaper2D(0, 0, r);      
        config.axis2D(0, 0,180, 1);        
          
        //坐标轴设定  
        var scaleX = 2*r, scaleY = 2*r;    
        var spaceX = 2, spaceY = 2;     
        var xS = -10, xE = 10;    
        var yS = -10, yE = 10;    
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');      
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');      
              
        var transform = new Transform();      
        //存放函数图像上的点  
        var a = [], b = [], c = [], d = [], e = [];    
            
        //需要显示的函数说明  
        var f1 = 'y=lnx+2x-6', f2 = 'y = logx', f3 = 'y=x', f4 = 'y=e^x',
			f5 = 'y = ln(x)';
        //函数描点  
        for (var x = xS; x <= xE; x+=0.3) {    
            if (x != 0) {  
                a.push([x, funTask(x)]);  




            }  
        }    

		//二分法求函数的零点
		//区间的最小值和最大值
		var minX = 0, maxX = 4;
		var y1 = y2 = 0;
		x = minX;
		y1 = funTask(x);
		x = maxX;
		y2 = funTask(x);
		
		//如果在给定区间上存在有零点
		if (y1 * y2 < 0) {
			var epsilon = 0.000001;
			while (Math.abs(y1-y2) > epsilon) {
			
				x = minX;
				y1 = funTask(x);
				x = maxX;
				y2 = funTask(x);
				
				
				x = (minX+maxX)/2;
				y = funTask(x);
				
				if (y * y1 < 0) {
					maxX = x;
				}
				else {
					minX = x;
				}
				

			}
			plot.setFillStyle('blue')
				.fillText('零点：x = '+ ((minX+maxX)/2).toFixed(3), -200, -60, 120);
		}
		</span>

这个题，由于[人叫板老师]的精度不够，所以取得的值并不如小伟的准确。

<span style="font-size:18px;">	if (1) {
		var r = 20;      
        config.setSector(1,1,1,1);        
        config.graphPaper2D(0, 0, r);      
        config.axis2D(0, 0,180, 1);        
          
        //坐标轴设定  
        var scaleX = 2*r, scaleY = 2*r;    
        var spaceX = 2, spaceY = 2;     
        var xS = -10, xE = 10;    
        var yS = -10, yE = 10;    
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');      
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');      
              
        var transform = new Transform();      
        //存放函数图像上的点  
        var a = [], b = [], c = [], d = [], e = [];    
            
        //需要显示的函数说明  
        var f1 = 'y=2^x+3x-7', f2 = 'y = logx', f3 = 'y=x', f4 = 'y=e^x',
			f5 = 'y = ln(x)';
        //函数描点  
        for (var x = xS; x <= xE; x+=0.3) {    
            if (x != 0) {  
                a.push([x, funTask(x)]);  




            }  
        }    

		//二分法求函数的零点
		//区间的最小值和最大值
		var minX = 0, maxX = 2;
		var y1 = y2 = 0;
		x = minX;
		y1 = funTask(x);
		x = maxX;
		y2 = funTask(x);
		
		//如果在给定区间上存在有零点
		if (y1 * y2 < 0) {
			var epsilon = 0.000001;
			while (Math.abs(y1-y2) > epsilon) {
			
				x = minX;
				y1 = funTask(x);
				x = maxX;
				y2 = funTask(x);
				
				
				x = (minX+maxX)/2;
				y = funTask(x);
				
				if (y * y1 < 0) {
					maxX = x;
				}
				else {
					minX = x;
				}
				

			}
			plot.setFillStyle('blue')
				.fillText('零点：x = '+ ((minX+maxX)/2).toFixed(3), -200, -60, 120);
		}
		
		//document.write(a+'<br/>');
        
		
        //存放临时数组  
        var tmp = [];  
            
        //显示变换  
        if (a.length > 0) {  
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);   
            //函数1  
            tmp = [].concat(a);      
            shape.pointDraw(tmp, 'red');      
            tmp = [].concat(a);      
            shape.multiLineDraw(tmp, 'pink');    
                
            plot.setFillStyle('red');  
            plot.fillText(f1, 100, -90, 200);    
        }  
          
        if (b.length > 0) {  
            b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);              
            //函数2  
            tmp = [].concat(b);      
            shape.pointDraw(tmp, 'blue');      
            tmp = [].concat(b);      
            shape.multiLineDraw(tmp, '#22CCFF');   
              
            plot.setFillStyle('blue');  
            plot.fillText(f2, 100, -120, 200);   
  
        }  
             
        if (c.length > 0) {  
            c = transform.scale(transform.translate(c, 0, 0), scaleX/spaceX, scaleY/spaceY);    
            tmp = [].concat(c);      
            shape.pointDraw(tmp, 'green');      
            tmp = [].concat(c);      
            shape.multiLineDraw(tmp, '#CCFF22');    
            plot.setFillStyle('green');    
            plot.fillText(f3, 100, -150, 200);   
        }  
          
        if (d.length > 0) {  
            d = transform.scale(transform.translate(d, 0, 0), scaleX/spaceX, scaleY/spaceY);    
            tmp = [].concat(d);      
            shape.pointDraw(tmp, 'orange');      
            tmp = [].concat(d);      
            shape.multiLineDraw(tmp, '#CC8800');    
            plot.setFillStyle('orange');    
            plot.fillText(f4, 100, -180, 200);   
        }  
  
        if (e.length > 0) {  
            e = transform.scale(transform.translate(e, 0, 0), scaleX/spaceX, scaleY/spaceY);    
            tmp = [].concat(e);      
            shape.pointDraw(tmp, 'purple');      
            tmp = [].concat(e);      
            shape.multiLineDraw(tmp, 'purple');    
            plot.setFillStyle('purple');    
            plot.fillText(f5, 100, -60, 200);   
        }  
  
    } 
	
	
	
}



function funTask(x) {
	return Math.pow(2, x)+3*x-7;
}</span>

<span style="font-size:18px;">function funTask(x) {
	return Math.pow(x, 3)+1.1*Math.pow(x, 2)+0.9*x-1.4;
}

function funTask(x) {
	return 3-Math.log(x)-x;
}
</span>

小伟决定来好好研究一下这个多项式拟合数据点。

这里有一组数据：

<span style="font-size:18px;">//一组点

	if (1) {
		var r = 20;      
        config.setSector(1,1,1,1);        
        config.graphPaper2D(0, 0, r);      
        config.axis2D(0, 0,180, 1);        
          
        //坐标轴设定  
        var scaleX = 2*r, scaleY = 2*r;    
        var spaceX = 2, spaceY = 2;     
        var xS = -10, xE = 10;    
        var yS = -10, yE = 10;    
		config.axisSpacing(xS, xE, spaceX, scaleX, 'X');      
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');   
		
		
		var X=[ 1 ,2  ,3 ,4 ,5 ,6], 
			Y=[ 2.5 ,3.51 ,4.45 ,5.52 ,6.47 ,7.51];
			
		var array = [];
		var size = X.length;
		
		for (var i = 0; i < size; i++) {
			array.push([X[i], Y[i]]);
		}
			
		var transform = new Transform();
		var tmp = [];
		
		array = transform.scale(transform.translate(array, 0, 0), scaleX/spaceX, scaleY/spaceY);    
		tmp = [].concat(array);      
		shape.pointDraw(tmp, 'green');      
		//tmp = [].concat(array);      
		//shape.multiLineDraw(tmp, '#CCFF22');    
			
	
	}</span>

经过这样的拟合计算：

<span style="font-size:18px;">#一次多项式拟合
def linefit(x , y):
    N = float(len(x))
    sx,sy,sxx,syy,sxy=0,0,0,0,0
    for i in range(0,int(N)):
        sx  += x[i]
        sy  += y[i]
        sxx += x[i]*x[i]
        syy += y[i]*y[i]
        sxy += x[i]*y[i]
    a = (sy*sx/N -sxy)/( sx*sx/N -sxx)
    b = (sy - a*sx)/N
    r = abs(sy*sx/N-sxy)/math.sqrt((sxx-sx*sx/N)*(syy-sy*sy/N))
    return a,b,r


def tmp():
    X=[ 1 ,2  ,3 ,4 ,5 ,6]
    Y=[ 2.5 ,3.51 ,4.45 ,5.52 ,6.47 ,7.51]
    a,b,r=linefit(X,Y)
    print("X=",X)
    print("Y=",Y)
    print("拟合结果: y = %10.5f x + %10.5f , r=%10.5f" % (a,b,r) )
	
>>> 
X= [1, 2, 3, 4, 5, 6]
Y= [2.5, 3.51, 4.45, 5.52, 6.47, 7.51]
拟合结果: y =    1.00000 x +    1.49333 , r=   0.99989</span>

来看一下拟合结果是否正确：

<span style="font-size:18px;">	if (1) {
		var r = 20;      
        config.setSector(1,1,1,1);        
        config.graphPaper2D(0, 0, r);      
        config.axis2D(0, 0,180, 1);        
          
        //坐标轴设定  
        var scaleX = 2*r, scaleY = 2*r;    
        var spaceX = 2, spaceY = 2;     
        var xS = -10, xE = 10;    
        var yS = -10, yE = 10;    
		config.axisSpacing(xS, xE, spaceX, scaleX, 'X');      
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');   
		
		
		var X=[ 1 ,2  ,3 ,4 ,5 ,6], 
			Y=[ 2.5 ,3.51 ,4.45 ,5.52 ,6.47 ,7.51];
			
		var array = [];
		var size = X.length;
		
		for (var i = 0; i < size; i++) {
			array.push([X[i], Y[i]]);
		}
			
		var transform = new Transform();
		var tmp = [];
		
  
			
	
		//绘制拟合曲线
		var a = [], b = [], c = [], d = [], e = [];    
            
        //需要显示的函数说明  
        var f1 = 'y=x+1.49333';
        //函数描点  
        for (var x = xS; x <= xE; x+=1) {    
            if (x != 0) {  
                a.push([x, funTask(x)]);  
			}  
        }  
		
		if (a.length > 0) {  
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);   
            //函数1  
            tmp = [].concat(a);      
            shape.pointDraw(tmp, 'red');      
            tmp = [].concat(a);      
            shape.multiLineDraw(tmp, 'pink');    
                
            plot.setFillStyle('red');  
            plot.fillText(f1, 100, -90, 200);    
        }  
		
		
		//原数据点
		array = transform.scale(transform.translate(array, 0, 0), scaleX/spaceX, scaleY/spaceY);    
		tmp = [].concat(array);      
		shape.pointDraw(tmp, 'green');      
		//tmp = [].concat(array);      
		//shape.multiLineDraw(tmp, '#CCFF22');  
		
		
	}
	
function funTask(x) {
	return x+1.49333;
}</span>

好像是没错。

来看看二次多项式拟合。

<span style="font-size:18px;">def polyfit(x, y, degree):
    import numpy;
    results = {}
    coeffs = numpy.polyfit(x, y, degree)
    results['polynomial'] = coeffs.tolist()

    # r-squared
    p = numpy.poly1d(coeffs)
    # fit values, and mean
    yhat = p(x)                         # or [p(z) for z in x]
    ybar = numpy.sum(y)/len(y)          # or sum(y)/len(y)
    ssreg = numpy.sum((yhat-ybar)**2)   # or sum([ (yihat - ybar)**2 for yihat in yhat])
    sstot = numpy.sum((y - ybar)**2)    # or sum([ (yi - ybar)**2 for yi in y])
    results['determination'] = ssreg / sstot #准确率
    return results


def tmp():
    x=[ 1 ,2  ,3 ,4 ,5 ,6]
    y=[ 2.5 ,3.51 ,4.45 ,5.52 ,6.47 ,7.2]
    z1 = polyfit(x, y, 2)
    print(z1);
	

>>> 
{'determination': 0.99893014166855909, 'polynomial': [-0.024285714285714542, 1.1257142857142872, 1.3699999999999986]}</span>

<span style="font-size:18px;">	if (1) {
		var r = 20;      
        config.setSector(1,1,1,1);        
        config.graphPaper2D(0, 0, r);      
        config.axis2D(0, 0,180, 1);        
          
        //坐标轴设定  
        var scaleX = 2*r, scaleY = 2*r;    
        var spaceX = 2, spaceY = 2;     
        var xS = -10, xE = 10;    
        var yS = -10, yE = 10;    
		config.axisSpacing(xS, xE, spaceX, scaleX, 'X');      
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');   
		
		
		var X=[ 1 ,2  ,3 ,4 ,5 ,6]
			Y=[ 2.5 ,3.51 ,4.45 ,5.52 ,6.47 ,7.2]
			
		var array = [];
		var size = X.length;
		
		for (var i = 0; i < size; i++) {
			array.push([X[i], Y[i]]);
		}
			
		var transform = new Transform();
		var tmp = [];
		
  
			
	
		//绘制拟合曲线
		var a = [], b = [], c = [], d = [], e = [];    
            
        //需要显示的函数说明  
        var f1 = 'y=-0.024x2+1.126x+1.370';
        //函数描点  
        for (var x = xS; x <= xE; x+=1) {    
            if (x != 0) {  
                a.push([x, funTask(x)]);  
			}  
        }  
		
		if (a.length > 0) {  
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);   
            //函数1  
            tmp = [].concat(a);      
            shape.pointDraw(tmp, 'red');      
            tmp = [].concat(a);      
            shape.multiLineDraw(tmp, 'pink');    
                
            plot.setFillStyle('red');  
            plot.fillText(f1, 100, -90, 200);    
        }  
		
		
		//原数据点
		array = transform.scale(transform.translate(array, 0, 0), scaleX/spaceX, scaleY/spaceY);    
		tmp = [].concat(array);      
		shape.pointDraw(tmp, 'green');      
		//tmp = [].concat(array);      
		//shape.multiLineDraw(tmp, '#CCFF22');  
		
		
	}
	</span>

这也不知道它怎么算的，明显有很大偏差啊。

这里顺便贴下对数拟合的代码，这个和小伟没任何关系，

因为不是小伟写的，一时也不好验证对错，先放着看看吧。

<span style="font-size:18px;">对数函数拟合-这个是最难的，baidu上都找不到，google了半天才找到的。指数、幂数拟合啥的，都用这个，把func改写一下就行
from scipy import log as log print pcov
import numpy
from scipy import log
from scipy.optimize import curve_fit

def func(x, a, b):
    y = a * log(x) + b
    return y

def polyfit(x, y, degree):
    results = {}
    #coeffs = numpy.polyfit(x, y, degree)
    popt, pcov = curve_fit(func, x, y)
    results['polynomial'] = popt

    # r-squared
    yhat = func(x ,popt[0] ,popt[1] )                         # or [p(z) for z in x]
    ybar = numpy.sum(y)/len(y)          # or sum(y)/len(y)
    ssreg = numpy.sum((yhat-ybar)**2)   # or sum([ (yihat - ybar)**2 for yihat in yhat])
    sstot = numpy.sum((y - ybar)**2)    # or sum([ (yi - ybar)**2 for yi in y])
    results['determination'] = ssreg / sstot

    return results


x=[ 1 ,2  ,3 ,4 ,5 ,6]
y=[ 2.5 ,3.51 ,4.45 ,5.52 ,6.47 ,7.51]
z1 = polyfit(x, y, 2)
print z1
</span>

本节到此结束，欲知后事如何，请看下回分解。