1. 简介

三对角线矩阵(Tridiagonal Matrix)，结构如公式(1)所示：

其中a1=0，cn=0。写成矩阵形式如(2):

常用的解法为Thomas algorithm，又称为The Tridiagonal matrix algorithm(TDMA). 它是一种高斯消元法的解法。分为两个阶段：向前消元（Forward Elimination）和回代（Back Substitution）。

• 向前消元（Forward Elimination）：

  c′i=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪cibicibi−aic′i−1;i=1;i=2,3,…,n−1(3)
  d′i=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪dibidi−aid′i−1bi−aic′i−1;i=1;i=2,3,…,n.(4)

• 回代（Back Substitution）：

  xn=d′nxi=d′i−c′ixi+1;i=n−1,n−2,…,1.(5)

2.代码

• 维基百科提供的C语言版本：

void solve_tridiagonal_in_place_destructive(float * restrict const x, const size_t X, const float * restrict const a, const float * restrict const b, float * restrict const c)
{/*solves Ax = v where A is a tridiagonal matrix consisting of vectors a, b, cx - initially contains the input vector v, and returns the solution x. indexed from 0 to X - 1 inclusiveX - number of equations (length of vector x)a - subdiagonal (means it is the diagonal below the main diagonal), indexed from 1 to X - 1 inclusiveb - the main diagonal, indexed from 0 to X - 1 inclusivec - superdiagonal (means it is the diagonal above the main diagonal), indexed from 0 to X - 2 inclusiveNote: contents of input vector c will be modified, making this a one-time-use function (scratch space can be allocated instead for this purpose to make it reusable)Note 2: We don't check for diagonal dominance, etc.; this is not guaranteed stable*//* index variable is an unsigned integer of same size as pointer */size_t ix;c[0] = c[0] / b[0];x[0] = x[0] / b[0];/* loop from 1 to X - 1 inclusive, performing the forward sweep */for (ix = 1; ix < X; ix++) {const float m = 1.0f / (b[ix] - a[ix] * c[ix - 1]);c[ix] = c[ix] * m;x[ix] = (x[ix] - a[ix] * x[ix - 1]) * m;}/* loop from X - 2 to 0 inclusive (safely testing loop condition for an unsigned integer), to perform the back substitution */for (ix = X - 1; ix-- > 0; )x[ix] = x[ix] - c[ix] * x[ix + 1];
}

• 本人基于Opencv的版本：

bool caltridiagonalMatrices( cv::Mat_<double> &input_a, cv::Mat_<double> &input_b, cv::Mat_<double> &input_c,cv::Mat_<double> &input_d,cv::Mat_<double> &output_x )
{/*solves Ax = v where A is a tridiagonal matrix consisting of vectors input_a, input_b, input_c, and v is a vector consisting of input_d.input_a - subdiagonal (means it is the diagonal below the main diagonal), indexed from 1 to X - 1 inclusiveinput_b - the main diagonal, indexed from 0 to X - 1 inclusiveinput_c - superdiagonal (means it is the diagonal above the main diagonal), indexed from 0 to X - 2 inclusiveinput_d - the input vector v, indexed from 0 to X - 1 inclusiveoutput_x - returns the solution x. indexed from 0 to X - 1 inclusive*//* the size of input_a is 1*n or n*1 */int rows = input_a.rows;int cols = input_a.cols;if ( ( rows == 1 && cols > rows ) || (cols == 1 && rows > cols ) ){const int count = ( rows > cols ? rows : cols ) - 1;output_x = cv::Mat_<double>::zeros(rows, cols);cv::Mat_<double> cCopy, dCopy;input_c.copyTo(cCopy);input_d.copyTo(dCopy);if ( input_b(0) != 0 ){cCopy(0) /= input_b(0);dCopy(0) /= input_b(0);}else{return false;}for ( int i=1; i < count; i++ ){double temp = input_b(i) - input_a(i) * cCopy(i-1);if ( temp == 0.0 ){return false;}cCopy(i) /= temp;dCopy(i) = ( dCopy(i) - dCopy(i-1)*input_a(i) ) / temp;}output_x(count) = dCopy(count);for ( int i=count-2; i > 0; i-- ){output_x(i) = dCopy(i) - cCopy(i)*output_x(i+1);}return true;}else{return false;}
}

参考文献：https://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm