Question

我想在Matlab中求解一个线性方程组。问题是这个系统通常会有一个非唯一的解决方案（所以Nullspace是非平凡的），这个系统依赖于参数beta（非零！）。因此，我想根据这个参数得到解决方案。 MATLAB能够做到这一点吗？我需要以什么方式输入方程式和参数以及我需要使用哪个命令，以便Matlab为我提供所有解决方案？

Answer 1

希望这会有所帮助。它并不意味着是最优的。它在八度音阶中进行了测试与matlab稍微不同的解析规则，我通常很好地保持共享八度和matlab的语法，但提供公平的警告。

    function x=solver(A,y,freeVars)
    %
    %  x=solver(A,y,freeVars)
    % 
    %  Solve system of equations Ax=y for x.
    %  Use elements of freeVars to fill undetermined ranks and produce
    %  a unique solution.
    %
    %  Typically this is of form 
    % 
    %   f_1( t_1 ) * x_1  +  f_2( t_1 ) * x_2 ...  + f_n( t_1 ) * x_n =  y_1
    %
    %   f_1( t_2 ) * x_1  +  f_2( t_2 ) * x_2 ...  + f_n( t_2 ) * x_n =  y_2
    %   .
    %   .
    %   .
    %   f_1( t_m ) * x_1  +  f_2( t_m ) * x_2 ...  + f_n( t_m ) * x_n =  y_m
    %
    %   A= [ f_1( t_1 ) , f_2( t_1 ) , ... f_n( t_1 ) ; 
    %        f_1( t_2 ) , f_2( t_2 ) , ... f_n( t_2 ) ;
    %        ...
    %        f_1( t_m ) , f_2( t_m ) , ... f_n( t_m ) ];
    %
    %  For example a first order linear fit would be
    %  f_1(t) = 1
    %  f_2(t) = t
    %
    %
    %  If the problem is overdetermined this would be a least squares problem 
    %  that is not going to be addressed here.
    %
    %  Assuming fully determined,  one solution would be
    %  Given:Ax=y
    %  [U,S,V]= svd(a)
    %  such that   U*S*V'*x = y
    %                S*V'*x = U'*y
    %  for fully determined case S is invertable.
    %  for less than fully determined case rank(S) < n, 
    %  Let [ S_r | 0 ]  represent the non-zero and zero columns of S.
    %  and [ V_r | 0 ]  represent the columns of V that are used vs. 
    %                   ones multiplied by zeros of S.
    %                [ S_r | 0 ] *  [ V_r |0 ]' * x  = [ U_r | 0 ]' *  y
    %
    %  V_r is in some sense a projection of your x coordinates into rank(S)
    %  subspace that is fully determined.  That portion can be solved
    %  but requires additional parameters to fully determine X.
    % 
    %                 x  =  V * [ inv(S_r)  U_r'  *  y ; alpha ]
    %
    % where alpha's are free parameters filling the extra degrees or freedom.
    %
    % The columns of V that aren't included in V_r are  (were temporarily 
    % temporarily replace by zeros determine which of the x parameters are 
    % impacted by each of the free parameters.
    %
    % Rather than use freevariables as I do here I presume one could set 
    % some x's that were influenced by those freevars to desired values 
    % and backsolve what values of free vars would produce those x's and 
    % then obtain values for the remaining undetermined x's from the computed
    % free vars.   
    %
    %
    [U,S,V]=svd(A)
    s=diag(S);
    %
    % Default rank tolerance taken from help page on rank.
    %
    r=sum(s>max(size(A)) * max(s)* eps )
    %
    % 
    U_r=U(:,1:r)
    S_r=S(1:r,1:r)
    %
    alpha = freeVars(1:(size(y,1)-r) ,1)
    %
    invS_r = diag(diag(S_r).^-1)
    x = V *  [ invS_r * U_r' * y  ; alpha ];
    %
    % aka:
    % x = V_r *  S_r^(-1) * U_r' *y   +   V_n * alpha

简单的测试用例

    % Fully determined case:  
    % mt+b = y   x=[b;m]=[1;2] evaluated at t=0, t=1
    % 
    t=[ 0 ; 1]
    %
    % A = [ 1 , t ]
    %
    A=[ ones(2,1) , t]
    %
    %
    xd=[ 1 ; 2 ]
    y = xd(1) + xd(2)* t

    x=solver(A,y,[1;2;3;4;5])
    xerr=xd-x
    yerr=A*x-y

    % under determined case:  
    % mt+b = y  w/ x=[b;m]=[1;2] evaluated at t=0, t=0
    % 
    t=[ 0 ; 0]
    %
    % A = [ 1 , t ]
    %
    A=[ ones(2,1) , t]
    %
    %
    xd=[ 1 ; 2 ]
    y = xd(1) + xd(2)* t

    x=solver(A,y,[1;2;3;4;5])
    xerr=xd-x
    yerr=A*x-y

Answer 2

Maxima [1]可以求解包含符号变量的方程式（如你所提到的beta），如果解决方案不是唯一的，它将引入虚拟变量，使得解决方案对所有虚拟变量值都有效。例如：

(%i5) solve ([3 * x + beta * y = 5, -6 * x - 2*beta * y = -10], [x, y]);
solve: dependent equations eliminated: (2)
                                %r2 beta - 5
(%o5)                   [[x = - ------------, y = %r2]]
                                     3

此处%r2是虚拟变量，解决方案适用于%r2的任何值。

然而，Maxima的符号解算器使用了大量内存，这可能会对它可以处理的问题的大小设置相对较低的限制。你有多少个方程式和多少个变量？也许你可以在这里发布方程组。

很抱歉，我不知道如何在Matlab中解决这个问题。

[1] http://maxima.sourceforge.net，http://sourceforge.net/p/maxima

求解Matlab中的方程组

2 个答案: