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/*
* Copyright (c) 2002-2006 Samit Basu
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*
*/
#include "LeastSquaresSolver.hpp"
#include "LAPACK.hpp"
#include <cstdlib>
#include <cstdio>
#include "MemPtr.hpp"
#include "Algorithms.hpp"
#define MSGBUFLEN 2048
template <typename T>
static void Tgelsy(int* M, int *N, int *NRHS, T* A, int *LDA,
T *B, int *LDB, int *JPVT, T* RCOND,
int *RANK, T *WORK, int* LWORK, int* INFO);
template <>
void Tgelsy(int* M, int *N, int *NRHS, double* A, int *LDA,
double *B, int *LDB, int *JPVT, double* RCOND,
int *RANK, double *WORK, int* LWORK, int* INFO) {
return dgelsy_(M,N,NRHS,A,LDA,B,LDB,JPVT,RCOND,
RANK,WORK,LWORK,INFO);
}
template <>
void Tgelsy(int* M, int *N, int *NRHS, float* A, int *LDA,
float *B, int *LDB, int *JPVT, float* RCOND,
int *RANK, float *WORK, int* LWORK, int* INFO) {
return sgelsy_(M,N,NRHS,A,LDA,B,LDB,JPVT,RCOND,
RANK,WORK,LWORK,INFO);
}
template <typename T>
static void Tgelsy(int* M, int *N, int *NRHS, T* A, int *LDA,
T *B, int *LDB, int *JPVT, T* RCOND,
int *RANK, T *WORK, int* LWORK, T* RWORK,
int* INFO);
template <>
void Tgelsy(int* M, int *N, int *NRHS, float* A, int *LDA,
float *B, int *LDB, int *JPVT, float* RCOND,
int *RANK, float *WORK, int* LWORK, float* RWORK,
int* INFO) {
return cgelsy_(M,N,NRHS,A,LDA,B,LDB,JPVT,RCOND,
RANK,WORK,LWORK,RWORK,INFO);
}
template <>
void Tgelsy(int* M, int *N, int *NRHS, double* A, int *LDA,
double *B, int *LDB, int *JPVT, double* RCOND,
int *RANK, double *WORK, int* LWORK, double* RWORK,
int* INFO) {
return zgelsy_(M,N,NRHS,A,LDA,B,LDB,JPVT,RCOND,
RANK,WORK,LWORK,RWORK,INFO);
}
/***************************************************************************
* Least-squares solver for double matrices
***************************************************************************/
/**
* Solve A * X = B in a least-squares sense, where A is m x n, and B is m x k.
* C is n x k.
*/
template <typename T>
static void realSolveLeastSq(int m, int n, int k, T *c, T *a, T *b) {
if ((m == 0) || (n == 0)) return;
// Here are the comments from the LAPACK routine used:
//SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
// WORK, LWORK, INFO )
//*
//* -- LAPACK driver routine (version 3.0) --
//* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
//* Courant Institute, Argonne National Lab, and Rice University
//* June 30, 1999
//*
//* .. Scalar Arguments ..
// INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
// DOUBLE PRECISION RCOND
//* ..
//* .. Array Arguments ..
// INTEGER JPVT( * )
// DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
//* ..
//*
//* Purpose
//* =======
//*
//* DGELSY computes the minimum-norm solution to a real linear least
//* squares problem:
//* minimize || A * X - B ||
//* using a complete orthogonal factorization of A. A is an M-by-N
//* matrix which may be rank-deficient.
//*
//* Several right hand side vectors b and solution vectors x can be
//* handled in a single call; they are stored as the columns of the
//* M-by-NRHS right hand side matrix B and the N-by-NRHS solution
//* matrix X.
//*
//* The routine first computes a QR factorization with column pivoting:
//* A * P = Q * [ R11 R12 ]
//* [ 0 R22 ]
//* with R11 defined as the largest leading submatrix whose estimated
//* condition number is less than 1/RCOND. The order of R11, RANK,
//* is the effective rank of A.
//*
//* Then, R22 is considered to be negligible, and R12 is annihilated
//* by orthogonal transformations from the right, arriving at the
//* complete orthogonal factorization:
//* A * P = Q * [ T11 0 ] * Z
//* [ 0 0 ]
//* The minimum-norm solution is then
//* X = P * Z' [ inv(T11)*Q1'*B ]
//* [ 0 ]
//* where Q1 consists of the first RANK columns of Q.
//*
//* This routine is basically identical to the original xGELSX except
//* three differences:
//* o The call to the subroutine xGEQPF has been substituted by the
//* the call to the subroutine xGEQP3. This subroutine is a Blas-3
//* version of the QR factorization with column pivoting.
//* o Matrix B (the right hand side) is updated with Blas-3.
//* o The permutation of matrix B (the right hand side) is faster and
//* more simple.
//*
//* Arguments
//* =========
//*
//* M (input) INTEGER
//* The number of rows of the matrix A. M >= 0.
int M = m;
//* N (input) INTEGER
//* The number of columns of the matrix A. N >= 0.
int N = n;
//* NRHS (input) INTEGER
//* The number of right hand sides, i.e., the number of
//* columns of matrices B and X. NRHS >= 0.
int NRHS = k;
//* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
//* On entry, the M-by-N matrix A.
//* On exit, A has been overwritten by details of its
//* complete orthogonal factorization.
T *A = a;
//* LDA (input) INTEGER
//* The leading dimension of the array A. LDA >= max(1,M).
int LDA = m;
//* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
//* On entry, the M-by-NRHS right hand side matrix B.
//* On exit, the N-by-NRHS solution matrix X.
int Bsize = (M > N) ? M : N;
// This passing convention requires that we copy our source matrix
// into the destination array with the appropriate padding.
MemBlock<T> B(Bsize*NRHS);
changeStride(&B,Bsize,b,m,m,NRHS);
//* LDB (input) INTEGER
//* The leading dimension of the array B. LDB >= max(1,M,N).
int LDB = Bsize;
//* JPVT (input/output) INTEGER array, dimension (N)
//* On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
//* to the front of AP, otherwise column i is a Free column.
//* On exit, if JPVT(i) = k, then the i-th column of AP
//* was the k-th column of A.
MemBlock<int> JPVT(N);
//* RCOND (input) DOUBLE PRECISION
//* RCOND is used to determine the effective rank of A, which
//* is defined as the order of the largest leading triangular
//* submatrix R11 in the QR factorization with pivoting of A,
//* whose estimated condition number < 1/RCOND.
T RCOND = lamch<T>();
//* RANK (output) INTEGER
//* The effective rank of A, i.e., the order of the submatrix
//* R11. This is the same as the order of the submatrix T11
//* in the complete orthogonal factorization of A.
int RANK;
//* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
//* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
T WORKSIZE;
//* LWORK (input) INTEGER
//* The dimension of the array WORK.
//* The unblocked strategy requires that:
//* LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
//* where MN = min( M, N ).
//* The block algorithm requires that:
//* LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
//* where NB is an upper bound on the blocksize returned
//* by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR,
//* and DORMRZ.
//*
//* If LWORK = -1, then a workspace query is assumed; the routine
//* only calculates the optimal size of the WORK array, returns
//* this value as the first entry of the WORK array, and no error
//* message related to LWORK is issued by XERBLA.
int LWORK;
//* INFO (output) INTEGER
//* = 0: successful exit
//* < 0: If INFO = -i, the i-th argument had an illegal value.
int INFO;
//* Further Details
//* ===============
//*
//* Based on contributions by
//* A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
//* E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
//* G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
//*
//* =====================================================================
//
LWORK = -1;
Tgelsy(&M, &N, &NRHS, A, &LDA, &B, &LDB, &JPVT, &RCOND,
&RANK, &WORKSIZE, &LWORK, &INFO);
LWORK = (int) WORKSIZE;
MemBlock<T> WORK(LWORK);
Tgelsy(&M, &N, &NRHS, A, &LDA, &B, &LDB, &JPVT, &RCOND,
&RANK, &WORK, &LWORK, &INFO);
// Check the rank...
if (M > N) {
// Problem should be overdetermined, rank should be N
if (RANK < N) {
WarningMessage(QString("Matrix is rank deficient to machine precision. RANK = %1\n").arg(RANK));
}
} else
// Problem should be underdetermined, rank should be M
if (RANK < M) {
WarningMessage(QString("Matrix is rank deficient to machine precision. RANK = %1\n").arg(RANK));
}
changeStride(c,n,&B,Bsize,n,k);
}
/***************************************************************************
* Least-squares solver for complex matrices
***************************************************************************/
/**
* Solve A * X = B in a least-squares sense, where A is m x n, and B is m x k.
* C is n x k.
*/
template <typename T>
static void complexSolveLeastSq(int m, int n, int k, T *c, T *a, T*b) {
if ((m == 0) || (n == 0)) return;
// SUBROUTINE ZGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
// $ WORK, LWORK, RWORK, INFO )
//*
//* -- LAPACK driver routine (version 3.0) --
//* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
//* Courant Institute, Argonne National Lab, and Rice University
//* June 30, 1999
//*
//* .. Scalar Arguments ..
// INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
// DOUBLE PRECISION RCOND
//* ..
//* .. Array Arguments ..
// INTEGER JPVT( * )
// DOUBLE PRECISION RWORK( * )
// COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
//* ..
//*
//* Purpose
//* =======
//*
//* ZGELSY computes the minimum-norm solution to a real linear least
//* squares problem:
//* minimize || A * X - B ||
//* using a complete orthogonal factorization of A. A is an M-by-N
//* matrix which may be rank-deficient.
//*
//* Several right hand side vectors b and solution vectors x can be
//* handled in a single call; they are stored as the columns of the
//* M-by-NRHS right hand side matrix B and the N-by-NRHS solution
//* matrix X.
//*
//* The routine first computes a QR factorization with column pivoting:
//* A * P = Q * [ R11 R12 ]
//* [ 0 R22 ]
//* with R11 defined as the largest leading submatrix whose estimated
//* condition number is less than 1/RCOND. The order of R11, RANK,
//* is the effective rank of A.
//*
//* Then, R22 is considered to be negligible, and R12 is annihilated
//* by orthogonal transformations from the right, arriving at the
//* complete orthogonal factorization:
//* A * P = Q * [ T11 0 ] * Z
//* [ 0 0 ]
//* The minimum-norm solution is then
//* X = P * Z' [ inv(T11)*Q1'*B ]
//* [ 0 ]
//* where Q1 consists of the first RANK columns of Q.
//*
//* This routine is basically identical to the original xGELSX except
//* three differences:
//* o The call to the subroutine xGEQPF has been substituted by the
//* the call to the subroutine xGEQP3. This subroutine is a Blas-3
//* version of the QR factorization with column pivoting.
//* o Matrix B (the right hand side) is updated with Blas-3.
//* o The permutation of matrix B (the right hand side) is faster and
//* more simple.
//*
//* Arguments
//* =========
//*
//* M (input) INTEGER
//* The number of rows of the matrix A. M >= 0.
int M = m;
//* N (input) INTEGER
//* The number of columns of the matrix A. N >= 0.
int N = n;
//* NRHS (input) INTEGER
//* The number of right hand sides, i.e., the number of
//* columns of matrices B and X. NRHS >= 0.
int NRHS = k;
//* A (input/output) COMPLEX*16 array, dimension (LDA,N)
//* On entry, the M-by-N matrix A.
//* On exit, A has been overwritten by details of its
//* complete orthogonal factorization.
T *A = a;
//* LDA (input) INTEGER
//* The leading dimension of the array A. LDA >= max(1,M).
int LDA = m;
//* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
//* On entry, the M-by-NRHS right hand side matrix B.
//* On exit, the N-by-NRHS solution matrix X.
int Bsize = (M > N) ? M : N;
// This passing convention requires that we copy our source matrix
// into the destination array with the appropriate padding.
MemBlock<T> B(Bsize*NRHS*2);
changeStride(&B,2*Bsize,b,2*m,2*m,NRHS);
//* LDB (input) INTEGER
//* The leading dimension of the array B. LDB >= max(1,M,N).
int LDB = Bsize;
//* JPVT (input/output) INTEGER array, dimension (N)
//* On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
//* to the front of AP, otherwise column i is a Free column.
//* On exit, if JPVT(i) = k, then the i-th column of AP
//* was the k-th column of A.
MemBlock<int> JPVT(N);
//* RCOND (input) DOUBLE PRECISION
//* RCOND is used to determine the effective rank of A, which
//* is defined as the order of the largest leading triangular
//* submatrix R11 in the QR factorization with pivoting of A,
//* whose estimated condition number < 1/RCOND.
T RCOND = lamch<T>();
//* RANK (output) INTEGER
//* The effective rank of A, i.e., the order of the submatrix
//* R11. This is the same as the order of the submatrix T11
//* in the complete orthogonal factorization of A.
int RANK;
//* WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
//* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
T WORKSIZE[2];
//* LWORK (input) INTEGER
//* The dimension of the array WORK.
//* The unblocked strategy requires that:
//* LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
//* where MN = min(M,N).
//* The block algorithm requires that:
//* LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
//* where NB is an upper bound on the blocksize returned
//* by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR,
//* and ZUNMRZ.
//*
//* If LWORK = -1, then a workspace query is assumed; the routine
//* only calculates the optimal size of the WORK array, returns
//* this value as the first entry of the WORK array, and no error
//* message related to LWORK is issued by XERBLA.
int LWORK;
//* RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
MemBlock<T> RWORK(2*N);
//* INFO (output) INTEGER
//* = 0: successful exit
//* < 0: If INFO = -i, the i-th argument had an illegal value.
int INFO;
//* Further Details
//* ===============
//*
//* Based on contributions by
//* A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
//* E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
//* G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
//*
//* =====================================================================
//
LWORK = -1;
Tgelsy(&M, &N, &NRHS, A, &LDA, &B, &LDB, &JPVT, &RCOND,
&RANK, WORKSIZE, &LWORK, &RWORK, &INFO);
LWORK = (int) WORKSIZE[0];
MemBlock<T> WORK(LWORK);
Tgelsy(&M, &N, &NRHS, A, &LDA, &B, &LDB, &JPVT, &RCOND,
&RANK, &WORK, &LWORK, &RWORK, &INFO);
// Check the rank...
if (M > N) {
// Problem should be overdetermined, rank should be N
if (RANK < N)
WarningMessage(QString("Matrix is rank deficient to machine precision. RANK = %1\n").arg(RANK));
} else
// Problem should be underderemined, rank should be M
if (RANK < M)
WarningMessage(QString("Matrix is rank deficient to machine precision. RANK = %1\n").arg(RANK));
changeStride(c,2*n,&B,2*Bsize,2*n,k);
}
template <typename T>
static Array SolveLeastSquaresReal(BasicArray<T> A, BasicArray<T> B) {
BasicArray<T> C(NTuple(A.cols(),B.cols()));
realSolveLeastSq(int(A.rows()),int(A.cols()),int(B.cols()),C.data(),A.data(),B.data());
return Array(C);
}
template <typename T>
static Array SolveLeastSquaresComplex(BasicArray<T> A, BasicArray<T> B) {
BasicArray<T> C(NTuple(2*A.cols(),B.cols()));
complexSolveLeastSq(int(A.rows()/2),int(A.cols()),int(B.cols()),C.data(),A.data(),B.data());
return Array(SplitReal<T>(C),SplitImag<T>(C));
}
// MxN * N x K = M x K
Array SolveLeastSquares(const Array & Ain, const Array &Bin) {
if (Ain.rows() != Bin.rows())
throw Exception("Solving Ax=B in a least squares sense requires the number of rows in A and B to be the same.");
DataClass via_type;
DataClass out_type;
ComputeTypes(Ain,Bin,via_type,out_type);
Array A = Ain.toClass(via_type);
Array B = Bin.toClass(via_type);
if (AnyNotFinite(A) || AnyNotFinite(B))
throw Exception("Solving Ax=b in a least squares sense does not currently support non-finite entries in A or B");
switch (A.dataClass()) {
default:
throw Exception("Unsupported data type for linear equation solver");
case Float:
if (A.allReal() && B.allReal())
return SolveLeastSquaresReal<float>(A.constReal<float>(),B.constReal<float>()).toClass(out_type);
else
return SolveLeastSquaresComplex<float>(A.fortran<float>(),B.fortran<float>()).toClass(out_type);
case Double:
if (A.allReal() && B.allReal())
return SolveLeastSquaresReal<double>(A.constReal<double>(),B.constReal<double>()).toClass(out_type);
else
return SolveLeastSquaresComplex<double>(A.fortran<double>(),B.fortran<double>()).toClass(out_type);
}
return Array();
}
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