三对角线性方程组(tridiagonal systems of equations)的求解

摘要：三对角线性方程组三对角线性方程组对于熟悉数值分析的同学来说，并不陌生，它经常出现在微分方程的数值求解和三次样条函数的插值问题中。

  三对角线性方程组,对于熟悉数值分析的同学来说，并不陌生，它经常出现在微分方程的数值求解和三次样条函数的插值问题中。三对角线性方程组可描述为以下方程组：
$$a_{i}x_{i-1}+b_{i}x_{i}+c_{i}x_{i+1}=d_{i}$$
其中$1leq i leq n, a_{1}=0, c_{n}=0.$ 以上方程组写成矩阵形式为$Ax=d$，即：

$$ {egin{bmatrix} {b_{1}}&{c_{1}}&{}&{}&{0} {a_{2}}&{b_{2}}&{c_{2}}&{}&{} {}&{a_{3}}&{b_{3}}&ddots &{} {}&{}&ddots &ddots &{c_{n-1}} {0}&&&{a_{n}}&{b_{n}} end{bmatrix}} {egin{bmatrix}{x_{1}}{x_{2}}{x_{3}}vdots {x_{n}}end{bmatrix}}={egin{bmatrix}{d_{1}}{d_{2}}{d_{3}}vdots {d_{n}}end{bmatrix}} $$

  三对角线性方程组的求解采用追赶法或者Thomas算法，它是Gauss消去法在三对角线性方程组这种特殊情形下的应用，因此，主要思想还是Gauss消去法，只是会更加简单些。我们将在下面的算法详述中给出该算法的具体求解过程。
  当然，该算法并不总是稳定的，但当系数矩阵$A$为严格对角占优矩阵（Strictly D iagonally Dominant, SDD）或对称正定矩阵（Symmetric Positive Definite, SPD）时，该算法稳定。对于不熟悉SDD或者SPD的读者，也不必担心，我们还会在我们的博客中介绍这类矩阵。现在，我们只要记住，该算法对于部分系数矩阵$A$是可以求解的。

  追赶法或者Thomas算法的具体步骤如下：

1.创建新系数$c_{i}^{*}$和$d_{i}^{*}$来代替原先的$a_{i},b_{i},c_{i}$,公式如下：

$$ c^{*}_i = left{ egin{array}{lr} frac{c_1}{b_1} & ; i = 1 frac{c_i}{b_i - a_i c^{*}_{i-1}} & ; i = 2,3,...,n-1 end{array} ight. d^{*}_i = left{ egin{array}{lr} frac{d_1}{b_1} & ; i = 1 frac{d_i- a_i d^{*}_{i-1}}{b_i - a_i c^{*}_{i-1}} & ; i = 2,3,...,n-1 end{array} ight. $$

2.改写原先的方程组$Ax=d$如下：

$$ egin{bmatrix} 1 & c^{*}_1 & 0 & 0 & ... & 0 0 & 1 & c^{*}_2 & 0 & ... & 0 0 & 0 & 1 & c^{*}_3 & 0 & 0 . & . & & & & . . & . & & & & . . & . & & & & c^{*}_{n-1} 0 & 0 & 0 & 0 & 0 & 1 end{bmatrix} egin{bmatrix} x_1 x_2 x_3 . . . x_k end{bmatrix} = egin{bmatrix} d^{*}_1 d^{*}_2 d^{*}_3 . . . d^{*}_n end{bmatrix} $$

3.计算解向量$x$，如下：
$$ x_n = d^{*}_n, qquad x_i = d^{*}_i - c^{*}_i x_{i+1}, qquad i = n-1, n-2, ... ,2,1$$

  以上算法得到的解向量$x$即为原方程$Ax=d$的解。
  下面，我们来证明该算法的正确性，只需要证明该算法保持原方程组的形式不变。
  首先，当$i=1$时，
$$1*x_{1}+c_{1}^{*}x_{2}=d_{1}^{*} Leftrightarrow 1*x_{1}+frac{c_{1}}{b_{1}}x_{2}=frac{d_{1}}{b_{1}}Leftrightarrow b_{1}*x_{1}+c_{1}x_{2}=d_{1}$$
  当$i>1$时，

$$ 1*x_{i}+c_{i}^{*}x_{i+1}=d_{i}^{*} Leftrightarrow 1*x_{i}+frac{c_{i}}{b_{i} - a_{i} c^{*}_{i-1}}x_{i+1}=frac{d_{i}- a_{i} d^{*}_{i-1}}{b_{i} - a_{i} c^{*}_{i-1}} Leftrightarrow (b_{i} - a_{i} c^{*}_{i-1})x_{i}+c_{i}x_{i+1}=d_{i}- a_{i} d^{*}_{i-1} $$

结合$a_{i}x_{i-1}+b_{i}x_{i}+c_{i}x_{i+1}=d_{i}$，只需要证明$x_{i-1}+c_{i-1}^{*}x_{i}=d_{i-1}^{*}$,而这已经在该算法的第（3）步的中的计算$x_{i-1}$中给出。证明完毕。

  我们将要求解的线性方程组如下：

$$ {egin{bmatrix} 4&1&{0}&{0}&{0} {1}&{4}&{1}&{0}&{0} {0}&{1}&{4}&{1}&{0} {0}&{0}&{1}&{4}&{1} {0}&{0}&{0}&{1}&{4} end{bmatrix}} {egin{bmatrix}{x_{1}}{x_{2}}{x_{3}}{x_{4}} {x_{5}}end{bmatrix}}={egin{bmatrix}{1.5 -132}end{bmatrix}} $$

  接下来，我们将用Python来实现该算法，函数为TDMA，输入参数为列表a,b,c,d, 输出为解向量x，代码如下：

# use Thomas Method to solve tridiagonal linear equation
# algorithm reference: https://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm

import numpy as np

# parameter: a,b,c,d are list-like of same length
# tridiagonal linear equation: Ax=d
# b: main diagonal of matrix A
# a: main diagonal below of matrix A
# c: main diagonal upper of matrix A
# d: Ax=d
# return: x(type=list), the solution of Ax=d
def TDMA(a,b,c,d):

    try:
        n = len(d)    # order of tridiagonal square matrix

        # use a,b,c to create matrix A, which is not necessary in the algorithm
        A = np.array([[0]*n]*n, dtype="float64")

        for i in range(n):
            A[i,i] = b[i]
            if i > 0:
                A[i, i-1] = a[i]
            if i < n-1:
                A[i, i+1] = c[i]

        # new list of modified coefficients
        c_1 = [0]*n
        d_1 = [0]*n

        for i in range(n):
            if not i:
                c_1[i] = c[i]/b[i]
                d_1[i] = d[i] / b[i]
            else:
                c_1[i] = c[i]/(b[i]-c_1[i-1]*a[i])
                d_1[i] = (d[i]-d_1[i-1]*a[i])/(b[i]-c_1[i-1] * a[i])

        # x: solution of Ax=d
        x = [0]*n

        for i in range(n-1, -1, -1):
            if i == n-1:
                x[i] = d_1[i]
            else:
                x[i] = d_1[i]-c_1[i]*x[i+1]

        x = [round(_, 4) for _ in x]

        return x

    except Exception as e:
        return e

def main():

    a = [0, 1, 1, 1, 1]
    b = [4, 4, 4, 4, 4]
    c = [1, 1, 1, 1, 0]
    d = [1, 0.5, -1, 3, 2]

    """
    a = [0, 2, 1, 3]
    b = [1, 1, 2, 1]
    c = [2, 3, 0.5, 0]
    d = [2, -1, 1, 3]
    """

    x = TDMA(a, b, c, d)
    print("The solution is %s"%x)

main()

运行该程序，输出结果为：

The solution is [0.2, 0.2, -0.5, 0.8, 0.3]

  本算法的Github地址为： https://github.com/percent4/N... .
  最后再次声明，追赶法或者Thomas算法并不是对所有的三对角矩阵都是有效的，只是部分三对角矩阵可行。

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