Opencv 三对角线矩阵(Tridiagonal Matrix)解法之(Thomas Algorithm)

1. 简介

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

aixi−1+bixi+cixx+1=di(1)

其中a1=0，cn=0。写成矩阵形式如(2):

⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢b1a20c1b2a3c2b3⋱⋱⋱cn−1an0bn⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎡⎣⎢⎢⎢⎢⎢⎢⎢x1x2x3⋮xn⎤⎦⎥⎥⎥⎥⎥⎥⎥=⎡⎣⎢⎢⎢⎢⎢⎢⎢d1d2d3⋮dn⎤⎦⎥⎥⎥⎥⎥⎥⎥(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