A = P L U
where P is a permutation matrix, L is lower trapezoidal with unit diagonal elements (lower triangular if n = m), and U is upper trapezoidal (upper triangular if n = m).
LA_LUDC is based on the following LAPACK routines:
The following example uses the LU decomposition on a given array, then determines the residual error of using the resulting lower and upper arrays to recompute the original array:
; Create a random array:
n = 20
seed = 12321
array = RANDOMN(seed, n, n, /RAN1)
; Compute LU decomposition.
aludc = array ; make a copy
LA_LUDC, aludc, index
; Extract the lower and upper triangular arrays.
l = IDENTITY(n)
u = FLTARR(n, n)
FOR j = 1,n - 1 DO l[0:j-1,j] = aludc[0:j-1,j]
FOR j=0,n - 1 DO u[j:*,j] = aludc[j:*,j]
; Reconstruct array, but with rows permuted.
arecon = l ## u
; Adjust from LAPACK back to IDL indexing.
Index = Index - 1
; Permute the array rows back into correct order.
; Note that we need to loop in reverse order.
temp = arecon[*,i]
arecon[*, i] = arecon[*,index[i]]
arecon[*, index[i]] = temp
When this program is compiled and run, IDL prints:
LA_LUDC error: 4.76837e-007
A named variable containing the real or complex array to decompose. This procedure returns Array as its LU decomposition.
An output vector with MIN(m, n) elements that records the row permutations which occurred as a result of partial pivoting. For 1 < j < MIN(m,n), row j of the matrix was interchanged with row Index[j].
Note: Row numbers within Index start at one rather than zero.
Set this keyword to use double-precision for computations and to return a double-precision (real or complex) result. Set DOUBLE = 0 to use single-precision for computations and to return a single-precision (real or complex) result. The default is /DOUBLE if Array is double precision, otherwise the default is DOUBLE = 0.
Set this keyword to a named variable that will contain the status of the computation. Possible values are:
- STATUS = 0: The computation was successful.
- STATUS > 0: One of the diagonal elements of U is zero. The STATUS value specifies which value along the diagonal (starting at one) is zero.
Note: If STATUS is not specified, any error messages will output to the screen.
Resources and References
For details see Anderson et al., LAPACK Users' Guide, 3rd ed., SIAM, 1999.