Linux下的數學工具Maxima 簡明教程（上）

    這個Blog裡曾經多次提到過超強數學軟體Mathematica，但目前為止我還沒發現它的Linux版，Wine似乎也沒有用。其實，在Linux下也有很多類似於Mathematica的數學軟體，其中Maxima是我用的最多的一個。這裡簡單介紹一下Maxima的各個函數供大家參考，也方便我自己今後查詢。

安裝：sudo apt-get install maxima maxima-share
運行：maxima
退出：quit();

基本運算
(%i1) 2+3;
(%o1)                                  5
(%i2) 5*6;
(%o2)                                 30
(%i3) %+2;
(%o3)                                 32
(%i4) %o1*%o3;
(%o4)                                 160
(%i5) 4/7+3/4;
                                      37
(%o5)                                 --
                                      28
(%i6) float(%);
(%o6)                          1.321428571428571
(%i7) 2^32;
(%o7)                             4294967296
(%i8) 30!;
(%o8)                  265252859812191058636308480000000
(%i9) float(sqrt(2));
(%o9)                          1.414213562373095

三角函數和對數函數
(%i1) float(sin(1));
(%o1)                           0.8414709848079
(%i2) sin(%pi/2);
(%o2)                                  1
(%i3) sin(%pi/2)+cos(%pi/3);
                                       3
(%o3)                                  -
                                       2
(%i4) float(sec(%pi/3)+csc(%pi/3));
(%o4)                          3.154700538379252
(%i5) log(1);
(%o5)                                  0
(%i6) float(log(10));
(%o6)                          2.302585092994046
(%i7) log(%e);
(%o7)                                  1
(%i8) log(2^a);
(%o8)                              log(2) a
(%i9) %e^log(2);
(%o9)                                 2

變數操作
(%i1) a^2-b^2;
                                     2    2
(%o1)                               a  - b
(%i2) a:3;
(%o2)                                  3
(%i3) a^2-b^2;
                                         2
(%o3)                               9 - b
(%i4) b:2;
(%o4)                                  2
(%i5) a^2-b^2;
(%o5)                                  5
(%i6) kill(a);
(%o6)                                done
(%i7) kill(b);
(%o7)                                done
(%i8) a^2-b^2;
                                     2    2
(%o8)                               a  - b

函數操作
(%i1) f(x):=x^2-1;
                                         2
(%o1)                           f(x) := x  - 1
(%i2) f(2);
(%o2)                                  3
(%i3) f(100);
(%o3)                                9999
(%i4) float(f(2/3));
(%o4)                         - 0.55555555555556
(%i5) a:4/5;
                                       4
(%o5)                                  -
                                       5
(%i6) f(a);
                                       9
(%o6)                                - --
                                       25

多項式運算（展開、合并、化簡和消元）
(%i1) expand((a+b)^3);
                            3        2      2      3
(%o1)                      b  + 3 a b  + 3 a  b + a
(%i2) factor(a^2-b^2);
(%o2)                          - (b - a) (b + a)
(%i3) ratsimp((x^2-1)/(x+1));
(%o3)                                x - 1
(%i4) eliminate([x^2+x*y+z=0,3*x+5*y+z=0,x-y-2*z^2=1],[y,z]);
                             4      3       2
(%o4)               [- x (8 x  - 2 x  + 19 x  - 50 x + 25)]

解方程
(%i1) solve(x^2-3*x+4/x=5,x);
                         sqrt(5) + 1      sqrt(5) - 1
(%o1)             [x = - -----------, x = -----------, x = 4]
                              2                2
(%i2) funcsolve(f(n)*(n+1)+2*n=1-f(n)/n,f(n));
                                      n (2 n - 1)
(%o2)                        f(n) = - -----------
                                       2
                                      n  + n + 1
(%i3) solve([x+3*y=10,1/x+x*y=4],[x,y]);
                              sqrt(69) - 9      4 sqrt(3) sqrt(23) - 34
(%o3) [[x = 1, y = 3], [x = - ------------, y = -----------------------],
                                   2            9 sqrt(3) sqrt(23) - 75
                                    sqrt(69) + 9      4 sqrt(3) sqrt(23) + 34
                               [x = ------------, y = -----------------------]]
                                         2            9 sqrt(3) sqrt(23) + 75
(%i4) solve(x^2+b*x+c=0,x);
                           2                       2
                     sqrt(b  - 4 c) + b      sqrt(b  - 4 c) - b
(%o4)         [x = - ------------------, x = ------------------]
                             2                       2
(%i5) find_root(x^x=2,x,1,2);
(%o5)                          1.559610469462369
(%i6) find_root(sin(x)=x/2,x,0.1,%pi);
(%o6)                          1.895494267033981

數論相關
(%i1) mod(100,7);
(%o1)                                  2
(%i2) primep(3214567);
(%o2)                                true
(%i3) next_prime(200);
(%o3)                                 211
(%i4) factor(1001);
(%o4)                               7 11 13
(%i5) factor(30!);
                        26  14  7  4   2   2
(%o5)                  2   3   5  7  11  13  17 19 23 29
(%i6) gcd(200,780);
(%o6)                                 20
(%i7) binomial(7,4);
(%o7)                                 35
(%i8) fib(7);
(%o8)                                 13

畫函數映像
(%i1) plot2d(x^3+2*x^2-3,[x,-2,2]);
*** X11 output driver not found, switching to dumb terminal!
*** If you want to use the X11 output, please install the gnuplot-x11 package

  14 ++-------+--------+--------+--------+-------+--------+--------+-------++
     +        +        +        +        +       +       x^3+2*x^2-3 $$$$$$ $
  12 ++                                                                    $+
     |                                                                    $ |
  10 ++                                                                  $ ++
     |                                                                  $   |
     |                                                                  $   |
   8 ++                                                                $   ++
     |                                                                $     |
   6 ++                                                             $$     ++
     |                                                             $$       |
   4 ++                                                          $$        ++
     |                                                          $$          |
   2 ++                                                        $$          ++
     |                                                      $$$             |
     |                                                     $$               |
   0 ++                                                 $$$                ++
     |                                               $$$$                   |
  -2 ++$$$$$$$$$$$$$$$$$$$$$$$$$$               $$$$$                      ++
     $$       +        +        $$$$$$$$$$$$$$$$ +        +        +        +
  -4 ++-------+--------+--------+--------+-------+--------+--------+-------++
    -2      -1.5      -1      -0.5       0      0.5       1       1.5       2

(%o1)

你可以通過安裝gnuplot-x11讓maxima在X上畫圖，安裝方法是：
sudo apt-get install gnuplot-x11
maxima也可以畫3D映像，比如執行下面代碼可以畫出sin(x)cos(y)的映像，我就不貼圖了，大家自己試試。
plot3d(sin(x)*cos(y),[x,-2,2],[y,-2,2]);

做人要厚道
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