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## Reversed solve after zgetrf ?

Hello,

I'm trying to implement a blocked LDLT factorization but i'm facing some problems with the blocked column update.

Is there any way to solve the equation X.A=B instead of A.X=B (zgetrs) after a LU decomposition of A with zgetrf ?

Pierre

1 Solution Employee
131 Views

Hi Pierre,

That's possible. First, after ZGETRF we have A=P*L*U (where P is row permutation storing in IPIV array).

So, X*A=B  <=>  X*P*L*U=B  <=>  ((X*P)*L)*U=B.  Let Y=X*P and  Z=Y*L.  In this case, you can solve:

(1) Z*U = B  - find Z using ZTRSM routine

(2) Y*L = Z - find Y using ZTRSM routine

(3) X*P = Y  <=> X=Y*P^-1  -  find X applying inverse permutation P.

Regards,

Konstantin

3 Replies Employee
132 Views

Hi Pierre,

That's possible. First, after ZGETRF we have A=P*L*U (where P is row permutation storing in IPIV array).

So, X*A=B  <=>  X*P*L*U=B  <=>  ((X*P)*L)*U=B.  Let Y=X*P and  Z=Y*L.  In this case, you can solve:

(1) Z*U = B  - find Z using ZTRSM routine

(2) Y*L = Z - find Y using ZTRSM routine

(3) X*P = Y  <=> X=Y*P^-1  -  find X applying inverse permutation P.

Regards,

Konstantin Beginner
131 Views

Hello Konstantin,

Thank you very much for your answer. I did not know i could use ztrsm on the output of zgetrf.

What i am supposed to do to apply the inverse permutation ? Is there a routine for that or what should I do ?

Thank you,

Pierre Black Belt
131 Views

The simplest solution, if you can rearrange your work so that you factorize AT instead of A, is to note that X A = B is the same as ATXT = BT. The last equation can be solved by a single call to ZGESV.

If, however, you wish to use the steps of #2, ...:

To undo the permutations, you can do the corresponding row interchanges after obtaining Y in order to obtain X, or you can form and use the inverse of the permutation vector when printing the solution. Here is an example (needs more testing for correctness!). For input data, you can use the file sgetrsx.d in the MKL examples/lapack/data directory.

```! Example program to illustrate solving X.A = B
!    1. Call ?GETRF to factorize A = P L U
!    2. Call ?TRSM  to solve Z U = B
!    3. Call ?TRSM  to solve Y L = Z
!    4. Apply inverse of P to retrieve X = Y inv(P)
!
! Ref.: software.intel.com/en-us/forums/intel-math-kernel-library/topic/702864
!
program strsmx
implicit none
integer nin, nout
parameter (nin=5, nout=6)
integer nmax, lda, nrhmax, ldb
parameter (nmax=8, lda=nmax, nrhmax=nmax, ldb=nrhmax)
!     .. Local Variables ..
integer i, ifail, info, j, k, n, nrhs
real a(lda, nmax), b(nrhmax, nmax), alpha
integer ipiv(nmax), iipiv(nmax)
!     .. Executable Statements ..
write (nout, *) 'STRSM Example Program Results'
!     Skip heading in data file
if (n<=nmax .and. nrhs<=nrhmax) then
!
!        Read A and B from data file
!
!
!        Factorize A
!
call sgetrf(n, n, a, lda, ipiv, info)
!
write (nout, *)
if (info/=0) then
write (nout, *) 'The matrix A is singular'
end if
!
!           Compute solution
!
alpha = 1.0
call strsm('R', 'U', 'N', 'N', n, n, alpha, a, lda, b, ldb)
call strsm('R', 'L', 'N', 'U', n, n, alpha, a, lda, b, ldb)
!
!        Form inverse of pivot array
!
iipiv(1:n) = (/ (i,i=1,n) /)
do i = n, 1, -1
j = ipiv(i)
if (j/=i) then
k = iipiv(i)
iipiv(i) = iipiv(j)
iipiv(j) = k
end if
end do
!
!           Print solution
!
do i = 1, n
write (*, '(10(2x,ES12.3))')(b(j,iipiv(i)), j=1, nrhs)
end do
end if
stop
!
end program strsmx``` 