Question

I think this must be simple but I can’t get it right…

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 m13 ]       [         m22 m23 ]       [             m33 ] 

is stored as

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

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) 

and

unsigned int column_index(i,M) 

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.

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

Answer 1

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; } 

EDIT: fixed row_index()