MatrixLUD
MatrixLUD [/B /CMF /PMAT /MIND /SUMP] srcWave
The MatrixLUD operation computes the LU factorization of a matrix. The general form of the factorization/decomposition is expressed in terms of matrix products:
M_Pt x srcWave = M_Lower x M_Upper
M_Pt, M_Lower and M_Upper are outputs created by MatrixLUD.
M_Pt is the transpose of the permutation matrix, M_Lower is a lower triangular matrix with 1's on the main diagonal and M_Upper is an upper triangular (or trapezoidal) matrix.
The MatrixLUD operation was substantially changed in Igor Pro 7.00. See the /B flag for information about backward compatibility.
Flags
| /B | This flag is provided for backward compatibility only; it is not compatible with any other flag. /B makes MatrixLUD behave as it did in Igor Pro 6. This flag is deprecated and will be removed in a future version of Igor. | |
| The input is restricted to a 2D real valued, single or double precision square matrix. The outputs (all double precision) are stored in the waves M_Upper, M_Lower and W_LUPermutation in the current data folder. | ||
| The W_LUPermutation output wave was needed for solving a linear system of equations using the back substitution routine, MatrixLUBkSub. For better computation methods see MatrixLinearSolve, MatrixLinearSolveTD and MatrixLLS. | ||
| /CMF | Uses Combined Matrix Format where the upper and lower matrix factors are combined into a single matrix saved in the wave M_LUFactors in the current data folder. The upper matrix factor is constructed from the main and from the upper diagonals of M_LUFactors. The lower matrix factor is constructed from the lower diagonals of M_LUFactors and setting the main diagonal to 1. | |
| /MIND | Finds the minimum magnitude diagonal element of M_Upper and store it in V_min. This is useful for investigating the behavior of the determinant of the matrix when it is close to being singular. | |
| /PMAT | Saves the transpose of the permutation matrix in a double precision wave M_Pt in the current data folder. Note that the permutation matrix is orthogonal and so the inverse of the matrix is equal to its transpose. | |
| /SUMP | Computes the sum of the phases of the elements on the main diagonal of M_Upper and store in the variable V_Sum. V_Sum is initialized to NaN and is set only if /SUMP is specified and M_Upper is complex. | |
Details
The input matrix srcWave is an MxN real or complex wave of single or double precision. Use MatrixLUDTD if your input is tri-diagonal.
The main results of the factorization are stored in the waves M_Lower, M_Upper and M_Pt. Alternatively the lower and upper factors can be combined and stored in the wave M_LUFactors (see /CMF). The waves M_Lower, M_Upper and M_LUFactors have the same data type as the input wave. M_Pt is always double precision.
When the input matrix srcWave is square (NxN), the resulting matrices have the same dimensions (NxN). You can reconstruct the input using the MatrixOp expression:
MatrixOp/O rA=(M_Pt^t) x (M_Lower x M_Upper)
If the input matrix is rectangular (NxM) the reconstruction depends on the size of N and M. If N<M:
MatrixOp/O rA=(M_Pt^t) x (subRange(M_lower,0,N-1,0,N-1) x M_Upper)
If N>M:
MatrixOp/O rA=(M_Pt^t) x M_lower x subRange(M_Upper,0,M-1,0,M-1)
The variable V_flag is set to zero if the operation succeeds and to 1 otherwise (e.g., if the input is singular). When you use the /B flag the polarity of the matrix is returned in the variable V_LUPolarity. The variables V_Sum and V_min are also set by some of the flag options above.
See Also
MatrixLUDTD, MatrixLUBkSub, MatrixLinearSolve, MatrixLinearSolveTD, MatrixLLS, MatrixOp, Matrix Math Operations