算法三角矩阵系数的索引号 [英] algorithm for index numbers of triangular matrix coefficients

问题描述

我想这一定是简单的，但我无法得到它的权利......

I think this must be simple but I can't get it right...

我有一个M×M个三角矩阵，其系数由行存储在向量，行。 例如：

I have an MxM triangular matrix, the coefficients of which are stored in a vector, row by row. For example:

M =   [ m00 m01 m02 m03 ] 
      [     m11 m12 m12 ]
      [         m22 m23 ]
      [             m33 ]

存储为

coef[ m00 m01 m02 m03 m11 m12 m13 m22 m23 m33 ]

现在我正在寻找一个非递归算法，让我为矩阵大小 M 和系数数组索引我

Now I'm looking for a non-recursive algorithm that gives me for matrix size M and coefficient array index i

unsigned int row_index(i,M)

和

unsigned int column_index(i,M)

矩阵元素的，它指的是。所以， row_index（9,4）== 3 ，与Column_Index（7,4）== 2 等，如果索引计数从零开始的。

of the matrix element that it refers to. So, row_index(9,4) == 3, column_index(7,4) == 2 etc. if the index counting is zero-based.

编辑：用迭代一些答复已经给出。有谁知道代数EX pressions？

Several replies using an iteration have been given. Does anyone know of algebraic expressions?

推荐答案

下面是一个代数（主要）解决方案：

Here's an algebraic (mostly) solution:

unsigned int row_index( unsigned int i, unsigned int M ){
    double m = M;
    double row = (-2*m - 1 + sqrt( (4*m*(m+1) - 8*(double)i - 7) )) / -2;
    if( row == (double)(int) row ) row -= 1;
    return (unsigned int) row;
}


unsigned int column_index( unsigned int i, unsigned int M ){
    unsigned int row = row_index( i, M);
    return  i - M * row + row*(row+1) / 2;
}

编辑：固定row_index（）

fixed row_index()

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