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, c

     x - initially contains the input vector v, and returns the solution x. indexed from 0 to X - 1 inclusive

     X - number of equations (length of vector x)

     a - subdiagonal (means it is the diagonal below the main diagonal), indexed from 1 to X - 1 inclusive

     b - the main diagonal, indexed from 0 to X - 1 inclusive

     c - superdiagonal (means it is the diagonal above the main diagonal), indexed from 0 to X - 2 inclusive

     Note: 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 inclusive

     input_b - the main diagonal, indexed from 0 to X - 1 inclusive

     input_c - superdiagonal (means it is the diagonal above the main diagonal), indexed from 0 to X - 2 inclusive

     input_d - the input vector v, indexed from 0 to X - 1 inclusive

     output_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