LU decomposition

From CFD-Wiki

Contents 1 Description 2 Algorithm 3 Important Considerations 4 References

Description

Consider the system of equations , where is an nonsingular matrix. may be decomposed into an lower triangular part and an upper triangular part that will lead us to a direct procedure for the solution of the original system. This decomposition procedure is quite useful when more than one right-hand side (more than one ) is to be used.

The algorithm is relatively straightforward - first, we determine the upper and lower triangular parts:

Then,

where . Once we solve the system

we will be able to find the solution to the original system by solving

The first solution is a foward substitution, while the second solution is a backward substitution. Both can be done efficiently once the factorization is available. The forward substitution may be expressed as

and the backward substitution process may be expressed as

LU decomposition essentially stores the operations of Gaussian elimination in "higher-level" form (see Golub and Van Loan), so repeated solutions using the same left-hand side are computed without repetition of operations that are independent of the right-hand side.

Algorithm

Add factorization here
Forward substitution

for k:=1 step until n do

for i:=1 step until k-1

end loop (i)

end loop (k)

Backward substitution

for k:=n stepdown until 1 do

for i:=k+1 step until n

end loop (i)

end loop (k)

Important Considerations

As with Gaussian elimination, LU decomposition is probably best used for relatively small, relatively non-sparse systems of equations (with small and non-sparse open to some interpretation). For larger and/or sparse problems, it would probably be best to either use an iterative method or use a direct solver package (e.g. DSCPACK) as opposed to writing one of your own.

If one has a single lefthand-side matrix and many right-hand side vectors, then LU decomposition would be a good solution procedure to consider. At the very least, it should be faster than solving each system separately with Gaussian elimination.

Here a Fortran code fragment for LU decomposition written by D. Partenov

do j = 1, n
 do i = 1, j
   if (i == 1) then
     a(i,j) = a(i, j)
   else
      suma = 0.0
     do k = 1, i-1
        suma = suma + a(i,k)*a(k,j)
     end do
     a(i,j) = a(i,j) - suma
   end if
 end do
 if ( j < n) then
   do s = 1, n-j
     i = j + s
     if (j == 1) then
       a(i,j) = a(i,j)/a(j,j)
     else
       suma = 0.0
       do k = 1, j-1
         suma = suma + a(i,k)*a(k,j)
       end do
       a(i,j) = (a(i,j) - suma)/a(j,j)
     end if
   end do
 end if
end do
y(1) = b(1)
do i = 2, n
 suma = 0.0
 do j = 1, i-1
   suma = suma + a(i,j)*y(j)
 end do
 y(i) = b(i) - suma
end do
i = n
j = n
x(i) = y(i)/a(i,j)
do s = 1, n -1
 i = n - s
 suma = 0.0
 do j = i+1, n
   suma = suma + a(i,j)*x(j)
 end do
 x(i) = (y(i) - suma)/a(i,i)
end do

References

• Golub and Van Loan (1996), Matrix Computations, 3rd edition, The Johns Hopkins University Press, Baltimore.
• Wikipedia article LU decomposition