算法：
为解决某类问题而设计的操作序列（非可执行的指令序列）
特点：有穷性、确定性、可行性、输入输出
1、遍试算法：
逻辑上：针对所有的可能的情况进行判断
形式上FOR中用IF示例：
- 韩信点兵
<span style="white-space:pre"> </span>using System;
class HanXin
{
static void Main()
{
for(int n=1;n<=105;n++)
if(n%3==2 && n%5==3 && n%7==5)
Console.WriteLine(n);
}
}
- 水仙花数：1^3+5^3+3^=153
<span style="white-space:pre"> </span>using Systen;
public class ShuiXianHua
{
public static void Main(string[] args)
{
for(int a=1;a<=9;a++)
for(int b=0;b<=1;b++)
for(int c=1;c<=9;c++)
if(a*a*a+b*b*b+c*c*c==100*a+10*b+c)
Console.WriteLine
}
}
运行结果：153、370、371、407
但人们目前最多知道11位的，随着位数增大，耗时增大，然后就算不出来了。
- 完全数：28=1+2+4+7+14
<span style="white-space:pre"> </span>using System;
class WanQuanShu
{
public static void Main(string[] args)
{
for(int n=1;n<=9999;n++)
if(n==divsum(n))
Console.WriteLine(n);
}
public static void divsum(int n)
{
int s=0;
for(int i=1;i<=n;i++)
if (n%i==0) s+=1;
return s;
}
}
- 相亲数:M的约数之和等于N的约数之和。算法和完全数有些类似
<span style="white-space:pre"> </span>using System；
class XiangQinShu
{
public static void Main(string[] args)
{
for(int n=1;n<=9999;n++)
{
int s=divsum(n)
if(n<s && divsum(s)==n)
Console.WriteLine(n+","+s);
}
}
public static void divsum(int n)
{
int s=0;
for(int i=1;i<=n;i++)
if (n%i==0) s+=1;
return s;
}
}
运行结果：220,284；1184,1210；2620,2924……
- 验证歌德巴猜想:任何一个偶数都能表示成两个质数之和。
<span style="white-space:pre"> </span>using System;
class GeDeBaCaiXiang
{
static void Main()
{
for(int n=6;n<=100;n+=2)
{
for(int a=3;a<n/2;a+=2)
int b=n-a;
if(IsPrime(a)&&IsPrime(b))
{
Console.WriteLine(n+"="+a+"+"+b);
break;
}
}
}
static bool IsPrime(int a)
{
for(int i=2;i<=a/2;i++)
{
if(s%i==0)
return false;
return true;
}
}
}
其他：百鸡问题、鸡兔同笼问题、百分比、佩尔方程
2、迭代：一步一步逐渐精确。
逻辑上：多次使用同一算法。
形式上：a=f(a)
示例：
- 求平方根
<span style="white-space:pre"> </span>using System;
public class Sqrt
{
public static void Main(string [] args){
Console.WriteLine( sqrt( 98) );
Console.WriteLine( Math.Sqrt(2.0) );
}
static double sqrt( double a ){
double x=1.0;
do{
Console.WriteLine( x + ", " + a/x );
x = ( x + a/x ) /2;
}while( Math.Abs(x*x-a)/a > 1e-6 );
return x;
}
}
- 倍边法求Pi
<span style="white-space:pre"> </span>using System;
class TestDebugPi
{
static void Main()
{
double a=1;
for(int n=1; n<=10; n++ )
{
a = Math.Sqrt(2 - Math.Sqrt(4 - a * a));
double pi = a * 3 * Math.Pow(2,n);
Console.WriteLine( pi );
}
Console.WriteLine( Math.PI );
}
}
其他：数字平方和、Mandelbrot集、Julia集
3、递归：分治思想，递归出口+递归主体
逻辑上：一个问题华为同样的问题
形式上：自己调用自己
示例：
- 求阶乘
<span style="white-space:pre"> </span>using System;
public class Fac
{
public static void Main(string [] args)
{
Console.WriteLine("Fac of 5 is " + fac( 5) );
}
static long fac( int n ){
if( n==0 || n==1) return 1;
else return fac(n-1) * n;
}
}
- 斐波那契数列
<span style="white-space:pre"> </span>using System;
public class Fibonacci
{
public static void Main(string [] args)
{
Console.WriteLine("Fibonacci(10) is " + fib(10) );
}
static long fib( int n ){
if( n==0 || n==1) return 1;
else return fib(n-1) + fib(n-2);
}
}
- Celay树
using System;
using System.Drawing;
using System.Collections;
using System.ComponentModel;
using System.Windows.Forms;
using System.Data;
public class Form1 : Form
{
public Form1()
{
this.AutoScaleBaseSize = new Size(6, 14);
this.ClientSize = new Size(600, 400);
this.Paint += new PaintEventHandler(this.Form1_Paint);
}
static void Main()
{
Application.Run(new Form1());
}
private void Form1_Paint(object sender, PaintEventArgs e)
{
graphics = e.Graphics ;
drawTree( 10, 200, 310, 100, -PI/2 );
}
private Graphics graphics;
const double PI = Math.PI;
double th1 = 40 * Math.PI / 180;
double th2 = 30 * Math.PI / 180;
double per1 = 0.6;
double per2 = 0.7;
void drawTree(int n,
double x0, double y0, double leng, double th)
{
if( n==0 ) return;
double x1 = x0 + leng * Math.Cos(th);
double y1 = y0 + leng * Math.Sin(th);
drawLine(x0, y0, x1, y1, n/2);
drawTree( n - 1, x1, y1, per1 * leng, th + th1 );
drawTree( n - 1, x1, y1, per2 * leng, th - th2 );
}
void drawLine( double x0, double y0, double x1, double y1, int width ){
graphics.DrawLine(
new Pen(Color.Blue, width ),
(int)x0, (int)y0, (int)x1, (int)y1 );
}
}
Celay树2：支随机持
using System;
using System.Drawing;
using System.Collections;
using System.ComponentModel;
using System.Windows.Forms;
using System.Data;
public class Form1 : Form
{
public Form1()
{
this.AutoScaleBaseSize = new Size(6, 14);
this.ClientSize = new Size(400, 400);
this.Paint += new PaintEventHandler(this.Form1_Paint);
this.Click += new EventHandler( this.Redraw );
}
static void Main()
{
Application.Run(new Form1());
}
private void Form1_Paint(object sender, PaintEventArgs e)
{
graphics = e.Graphics ;
drawTree( 10, 200, 380, 100, -PI/2 );
}
private void Redraw(object sender, EventArgs e)
{
this.Invalidate();
}
private Graphics graphics;
const double PI = Math.PI;
double th1 = 35 * Math.PI / 180;
double th2 = 25 * Math.PI / 180;
double per1 = 0.6;
double per2 = 0.7;
Random rnd = new Random();
double rand()
{
return rnd.NextDouble();
}
void drawTree(int n,
double x0, double y0, double leng, double th)
{
if( n==0 ) return;
double x1 = x0 + leng * Math.Cos(th);
double y1 = y0 + leng * Math.Sin(th);
drawLine(x0, y0, x1, y1);
drawTree( n - 1, x1, y1, per1 * leng * (0.5+rand()), th + th1* (0.5+rand()) );
drawTree( n - 1, x1, y1, per2 * leng * (0.4+rand()), th - th2* (0.5+rand()) );
if(rand()>0.6)
drawTree( n - 1, x1, y1, per2 * leng * (0.4+rand()), th - th2* (0.5+rand()) );
}
void drawLine( double x0, double y0, double x1, double y1 ){
graphics.DrawLine(
Pens.Blue,
(int)x0, (int)y0, (int)x1, (int)y1 );
}
}
其他：Koch分形集
小结：
遍试：FOR中用IF
迭代：WHILE中a=F(a)；
递归：F(n)中用F(n-1);
本文在学习唐大仕老师在网易云课堂中C#公开课后编辑而成，再此表示感谢！