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Hello,

I'm trying to implement a blocked LDL^{T} 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 ?

Thank you in advance,

Pierre

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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

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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

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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

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The simplest solution, if you can rearrange your work so that you factorize A^{T} instead of A, is to note that X A = B is the same as A^{T}X^{T} = B^{T}. 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 read (nin, *) read (nin, *) n, nrhs if (n<=nmax .and. nrhs<=nrhmax) then ! ! Read A and B from data file ! read (nin, *)((a(i,j),j=1,n), i=1, n) read (nin, *)((b(j,i),j=1,nrhs), i=1, n) ! ! 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

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