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/*
* Copyright (c) 2011, Michael Lehn
*
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1) Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2) Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in
* the documentation and/or other materials provided with the
* distribution.
* 3) Neither the name of the FLENS development group nor the names of
* its contributors may be used to endorse or promote products derived
* from this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
* A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
/* Based on
*
SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
$ X, LDX, FERR, BERR, WORK, IWORK, INFO )
SUBROUTINE ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
$ X, LDX, FERR, BERR, WORK, RWORK, INFO )
*
* -- LAPACK routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*/
#ifndef FLENS_LAPACK_GE_RFS_TCC
#define FLENS_LAPACK_GE_RFS_TCC 1
#include <flens/blas/blas.h>
#include <flens/lapack/lapack.h>
namespace flens { namespace lapack {
//== generic lapack implementation =============================================
namespace generic {
//-- (ge)rfs [real variant] ----------------------------------------------------
template <typename MA, typename MAF, typename VPIV, typename MB, typename MX,
typename VFERR, typename VBERR, typename VWORK, typename VIWORK>
void
rfs_impl(Transpose trans,
const GeMatrix<MA> &A,
const GeMatrix<MAF> &AF,
const DenseVector<VPIV> &piv,
const GeMatrix<MB> &B,
GeMatrix<MX> &X,
DenseVector<VFERR> &fErr,
DenseVector<VBERR> &bErr,
DenseVector<VWORK> &work,
DenseVector<VIWORK> &iwork)
{
using std::abs;
typedef typename GeMatrix<MA>::ElementType ElementType;
typedef typename GeMatrix<MA>::IndexType IndexType;
const IndexType itMax = 5;
const ElementType Zero(0), One(1), Two(2), Three(3);
const Underscore<IndexType> _;
const IndexType n = B.numRows();
const IndexType nRhs = B.numCols();
//
// Local Arrays
//
IndexType iSaveData[3] = {0, 0, 0};
DenseVectorView<IndexType>
iSave = typename DenseVectorView<IndexType>::Engine(3, iSaveData, 1);
//
// Quick return if possible
//
if (n==0 || nRhs==0) {
fErr = Zero;
bErr = Zero;
return;
}
const Transpose transT = (trans==NoTrans) ? Trans : NoTrans;
//
// NZ = maximum number of nonzero elements in each row of A, plus 1
//
const IndexType nz = n + 1;
const ElementType eps = lamch<ElementType>(Eps);
const ElementType safeMin = lamch<ElementType>(SafeMin);
const ElementType safe1 = nz * safeMin;
const ElementType safe2 = safe1 / eps;
auto work1 = work(_(1,n));
auto work2 = work(_(n+1,2*n));
auto work3 = work(_(2*n+1,3*n));
//
// Do for each right hand side
//
for (IndexType j=1; j<=nRhs; ++j) {
IndexType count = 1;
ElementType lastRes = Three;
RETRY:
//
// Loop until stopping criterion is satisfied.
//
// Compute residual R = B - op(A) * X,
// where op(A) = A, A**T, or A**H, depending on TRANS.
//
work2 = B(_,j);
blas::mv(trans, -One, A, X(_,j), One, work2);
//
// Compute componentwise relative backward error from formula
//
// max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
//
// where abs(Z) is the componentwise absolute value of the matrix
// or vector Z. If the i-th component of the denominator is less
// than SAFE2, then SAFE1 is added to the i-th components of the
// numerator and denominator before dividing.
//
for (IndexType i=1; i<=n; ++i) {
work(i) = abs(B(i,j));
}
//
// Compute abs(op(A))*abs(X) + abs(B).
//
if (trans==NoTrans) {
for (IndexType k=1; k<=n; ++k) {
const ElementType xk = abs(X(k,j));
for (IndexType i=1; i<=n; ++i) {
work(i) += abs(A(i,k)) * xk;
}
}
} else {
for (IndexType k=1; k<=n; ++k) {
ElementType s = Zero;
for (IndexType i=1; i<=n; ++i) {
s += abs(A(i,k)) * abs(X(i,j));
}
work1(k) += s;
}
}
ElementType s = Zero;
for (IndexType i=1; i<=n; ++i) {
if (work1(i)>safe2) {
s = max(s, abs(work2(i))/work1(i));
} else {
s = max(s, (abs(work2(i))+safe1)/(work1(i)+safe1));
}
}
bErr(j) = s;
//
// Test stopping criterion. Continue iterating if
// 1) The residual BERR(J) is larger than machine epsilon, and
// 2) BERR(J) decreased by at least a factor of 2 during the
// last iteration, and
// 3) At most ITMAX iterations tried.
//
if (bErr(j)>eps && Two*bErr(j)<=lastRes && count<=itMax) {
//
// Update solution and try again.
//
trs(trans, AF, piv, work2);
X(_,j) += work2;
lastRes = bErr(j);
++count;
goto RETRY;
}
//
// Bound error from formula
//
// norm(X - XTRUE) / norm(X) .le. FERR =
// norm( abs(inv(op(A)))*
// ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
//
// where
// norm(Z) is the magnitude of the largest component of Z
// inv(op(A)) is the inverse of op(A)
// abs(Z) is the componentwise absolute value of the matrix or
// vector Z
// NZ is the maximum number of nonzeros in any row of A, plus 1
// EPS is machine epsilon
//
// The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
// is incremented by SAFE1 if the i-th component of
// abs(op(A))*abs(X) + abs(B) is less than SAFE2.
//
// Use DLACN2 to estimate the infinity-norm of the matrix
// inv(op(A)) * diag(W),
// where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
//
for (IndexType i=1; i<=n; ++i) {
if (work(i)>safe2) {
work(i) = abs(work2(i)) + nz*eps*work1(i);
} else {
work(i) = abs(work2(i)) + nz*eps*work1(i) + safe1;
}
}
IndexType kase = 0;
while (true) {
lacn2(work3, work2, iwork, fErr(j), kase, iSave);
if (kase==0) {
break;
}
if (kase==1) {
//
// Multiply by diag(W)*inv(op(A)**T).
//
trs(transT, AF, piv, work2);
for (IndexType i=1; i<=n; ++i) {
work2(i) *= work1(i);
}
} else {
//
// Multiply by inv(op(A))*diag(W).
//
for (IndexType i=1; i<=n; ++i) {
work2(i) *= work1(i);
}
trs(trans, AF, piv, work2);
}
}
//
// Normalize error.
//
lastRes = Zero;
for (IndexType i=1; i<=n; ++i) {
lastRes = max(lastRes, abs(X(i,j)));
}
if (lastRes!=Zero) {
fErr(j) /= lastRes;
}
}
}
} // namespace generic
//== interface for native lapack ===============================================
#ifdef USE_CXXLAPACK
namespace external {
//-- (ge)rfs [real and complex variant] ----------------------------------------
template <typename MA, typename MAF, typename VPIV, typename MB, typename MX,
typename VFERR, typename VBERR, typename VWORK, typename VWORK2>
void
rfs_impl(Transpose trans,
const GeMatrix<MA> &A,
const GeMatrix<MAF> &AF,
const DenseVector<VPIV> &piv,
const GeMatrix<MB> &B,
GeMatrix<MX> &X,
DenseVector<VFERR> &fErr,
DenseVector<VBERR> &bErr,
DenseVector<VWORK> &work,
DenseVector<VWORK2> &work2)
{
typedef typename GeMatrix<MA>::IndexType IndexType;
IndexType info = cxxlapack::gerfs<IndexType>(getF77Char(trans),
B.numRows(),
B.numCols(),
A.data(),
A.leadingDimension(),
AF.data(),
AF.leadingDimension(),
piv.data(),
B.data(),
B.leadingDimension(),
X.data(),
X.leadingDimension(),
fErr.data(),
bErr.data(),
work.data(),
work2.data());
ASSERT(info==0);
FAKE_USE_NDEBUG(info);
}
} // namespace external
#endif // USE_CXXLAPACK
//== public interface ==========================================================
//-- (ge)rfs [real variant] ----------------------------------------------------
template <typename MA, typename MAF, typename VPIV, typename MB, typename MX,
typename VFERR, typename VBERR, typename VWORK, typename VIWORK>
typename RestrictTo<IsRealGeMatrix<MA>::value
&& IsRealGeMatrix<MAF>::value
&& IsIntegerDenseVector<VPIV>::value
&& IsRealGeMatrix<MB>::value
&& IsRealGeMatrix<MX>::value
&& IsRealDenseVector<VFERR>::value
&& IsRealDenseVector<VBERR>::value
&& IsRealDenseVector<VWORK>::value
&& IsIntegerDenseVector<VIWORK>::value,
void>::Type
rfs(Transpose trans,
const MA &A,
const MAF &AF,
const VPIV &piv,
const MB &B,
MX &&X,
VFERR &&fErr,
VBERR &&bErr,
VWORK &&work,
VIWORK &&iwork)
{
LAPACK_DEBUG_OUT("(ge)rfs [real]");
//
// Remove references from rvalue types
//
# ifndef NDEBUG
typedef typename RemoveRef<MA>::Type MatrixA;
typedef typename MatrixA::IndexType IndexType;
# endif
# ifdef CHECK_CXXLAPACK
typedef typename RemoveRef<MX>::Type MatrixX;
typedef typename RemoveRef<VFERR>::Type VectorFErr;
typedef typename RemoveRef<VBERR>::Type VectorBErr;
typedef typename RemoveRef<VWORK>::Type VectorWork;
typedef typename RemoveRef<VIWORK>::Type VectorIWork;
# endif
//
// Test the input parameters
//
# ifndef NDEBUG
ASSERT(A.firstRow()==1);
ASSERT(A.firstCol()==1);
ASSERT(A.numRows()==A.numCols());
const IndexType n = A.numRows();
ASSERT(AF.firstRow()==1);
ASSERT(AF.firstCol()==1);
ASSERT(AF.numRows()==n);
ASSERT(AF.numCols()==n);
ASSERT(piv.firstIndex()==1);
ASSERT(piv.length()==n);
ASSERT(B.firstRow()==1);
ASSERT(B.firstCol()==1);
ASSERT(B.numRows()==n);
const IndexType nRhs = B.numCols();
ASSERT(X.firstRow()==1);
ASSERT(X.firstCol()==1);
ASSERT(X.numRows()==n);
ASSERT(X.numCols()==nRhs);
ASSERT(fErr.firstIndex()==1);
ASSERT(fErr.length()==nRhs);
ASSERT(bErr.firstIndex()==1);
ASSERT(bErr.length()==nRhs);
ASSERT(work.firstIndex()==1);
ASSERT(work.length()==3*n);
ASSERT(iwork.firstIndex()==1);
ASSERT(iwork.length()==n);
# endif
//
// Make copies of output arguments
//
# ifdef CHECK_CXXLAPACK
typename MatrixX::NoView X_org = X;
typename VectorFErr::NoView fErr_org = fErr;
typename VectorBErr::NoView bErr_org = bErr;
typename VectorWork::NoView work_org = work;
typename VectorIWork::NoView iwork_org = iwork;
# endif
//
// Call implementation
//
LAPACK_SELECT::rfs_impl(trans, A, AF, piv, B, X, fErr, bErr, work, iwork);
# ifdef CHECK_CXXLAPACK
//
// Compare results
//
typename MatrixX::NoView X_generic = X;
typename VectorFErr::NoView fErr_generic = fErr;
typename VectorBErr::NoView bErr_generic = bErr;
typename VectorWork::NoView work_generic = work;
typename VectorIWork::NoView iwork_generic = iwork;
X = X_org;
fErr = fErr_org;
bErr = bErr_org;
work = work_org;
iwork = iwork_org;
external::rfs_impl(trans, A, AF, piv, B, X, fErr, bErr, work, iwork);
bool failed = false;
if (! isIdentical(X_generic, X, "X_generic", "X")) {
std::cerr << "CXXLAPACK: X_generic = " << X_generic << std::endl;
std::cerr << "F77LAPACK: X = " << X << std::endl;
failed = true;
}
if (! isIdentical(fErr_generic, fErr, "fErr_generic", "fErr")) {
std::cerr << "CXXLAPACK: fErr_generic = " << fErr_generic << std::endl;
std::cerr << "F77LAPACK: fErr = " << fErr << std::endl;
failed = true;
}
if (! isIdentical(bErr_generic, bErr, "bErr_generic", "bErr")) {
std::cerr << "CXXLAPACK: bErr_generic = " << bErr_generic << std::endl;
std::cerr << "F77LAPACK: bErr = " << bErr << std::endl;
failed = true;
}
if (! isIdentical(work_generic, work, "work_generic", "work")) {
std::cerr << "CXXLAPACK: work_generic = " << work_generic << std::endl;
std::cerr << "F77LAPACK: work = " << work << std::endl;
failed = true;
}
if (! isIdentical(iwork_generic, iwork, "iwork_generic", "iwork")) {
std::cerr << "CXXLAPACK: iwork_generic = "
<< iwork_generic << std::endl;
std::cerr << "F77LAPACK: iwork = " << iwork << std::endl;
failed = true;
}
if (failed) {
ASSERT(0);
}
# endif
}
//-- (ge)rfs [complex variant] -------------------------------------------------
#ifdef USE_CXXLAPACK
template <typename MA, typename MAF, typename VPIV, typename MB, typename MX,
typename VFERR, typename VBERR, typename VWORK, typename VRWORK>
typename RestrictTo<IsComplexGeMatrix<MA>::value
&& IsComplexGeMatrix<MAF>::value
&& IsIntegerDenseVector<VPIV>::value
&& IsComplexGeMatrix<MB>::value
&& IsComplexGeMatrix<MX>::value
&& IsRealDenseVector<VFERR>::value
&& IsRealDenseVector<VBERR>::value
&& IsComplexDenseVector<VWORK>::value
&& IsRealDenseVector<VRWORK>::value,
void>::Type
rfs(Transpose trans,
const MA &A,
const MAF &AF,
const VPIV &piv,
const MB &B,
MX &&X,
VFERR &&fErr,
VBERR &&bErr,
VWORK &&work,
VRWORK &&rwork)
{
LAPACK_DEBUG_OUT("(ge)rfs [complex]");
# ifndef NDEBUG
//
// Remove references from rvalue types
//
typedef typename RemoveRef<MA>::Type MatrixA;
typedef typename MatrixA::IndexType IndexType;
//
// Test the input parameters
//
ASSERT(A.firstRow()==1);
ASSERT(A.firstCol()==1);
ASSERT(A.numRows()==A.numCols());
const IndexType n = A.numRows();
ASSERT(AF.firstRow()==1);
ASSERT(AF.firstCol()==1);
ASSERT(AF.numRows()==n);
ASSERT(AF.numCols()==n);
ASSERT(piv.firstIndex()==1);
ASSERT(piv.length()==n);
ASSERT(B.firstRow()==1);
ASSERT(B.firstCol()==1);
ASSERT(B.numRows()==n);
const IndexType nRhs = B.numCols();
ASSERT(X.firstRow()==1);
ASSERT(X.firstCol()==1);
ASSERT(X.numRows()==n);
ASSERT(X.numCols()==nRhs);
ASSERT(fErr.firstIndex()==1);
ASSERT(fErr.length()==nRhs);
ASSERT(bErr.firstIndex()==1);
ASSERT(bErr.length()==nRhs);
ASSERT(work.firstIndex()==1);
ASSERT(work.length()==2*n);
ASSERT(rwork.firstIndex()==1);
ASSERT(rwork.length()==n);
# endif
//
// Call implementation
//
external::rfs_impl(trans, A, AF, piv, B, X, fErr, bErr, work, rwork);
}
#endif // USE_CXXLAPACK
} } // namespace lapack, flens
#endif // FLENS_LAPACK_GE_RFS_TCC
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