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qr_solve.cpp
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/**
* Marlin 3D Printer Firmware
* Copyright (C) 2016 MarlinFirmware [https://github.com/MarlinFirmware/Marlin]
*
* Based on Sprinter and grbl.
* Copyright (C) 2011 Camiel Gubbels / Erik van der Zalm
*
* 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 3 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, see <http://www.gnu.org/licenses/>.
*
*/
#include "qr_solve.h"
#if ENABLED(AUTO_BED_LEVELING_GRID)
#include <stdlib.h>
#include <math.h>
//# include "r8lib.h"
int i4_min(int i1, int i2)
/******************************************************************************/
/**
Purpose:
I4_MIN returns the smaller of two I4's.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
29 August 2006
Author:
John Burkardt
Parameters:
Input, int I1, I2, two integers to be compared.
Output, int I4_MIN, the smaller of I1 and I2.
*/
{
return (i1 < i2) ? i1 : i2;
}
double r8_epsilon(void)
/******************************************************************************/
/**
Purpose:
R8_EPSILON returns the R8 round off unit.
Discussion:
R8_EPSILON is a number R which is a power of 2 with the property that,
to the precision of the computer's arithmetic,
1 < 1 + R
but
1 = ( 1 + R / 2 )
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
01 September 2012
Author:
John Burkardt
Parameters:
Output, double R8_EPSILON, the R8 round-off unit.
*/
{
const double value = 2.220446049250313E-016;
return value;
}
double r8_max(double x, double y)
/******************************************************************************/
/**
Purpose:
R8_MAX returns the maximum of two R8's.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
07 May 2006
Author:
John Burkardt
Parameters:
Input, double X, Y, the quantities to compare.
Output, double R8_MAX, the maximum of X and Y.
*/
{
return (y < x) ? x : y;
}
double r8_abs(double x)
/******************************************************************************/
/**
Purpose:
R8_ABS returns the absolute value of an R8.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
07 May 2006
Author:
John Burkardt
Parameters:
Input, double X, the quantity whose absolute value is desired.
Output, double R8_ABS, the absolute value of X.
*/
{
return (x < 0.0) ? -x : x;
}
double r8_sign(double x)
/******************************************************************************/
/**
Purpose:
R8_SIGN returns the sign of an R8.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
08 May 2006
Author:
John Burkardt
Parameters:
Input, double X, the number whose sign is desired.
Output, double R8_SIGN, the sign of X.
*/
{
return (x < 0.0) ? -1.0 : 1.0;
}
double r8mat_amax(int m, int n, double a[])
/******************************************************************************/
/**
Purpose:
R8MAT_AMAX returns the maximum absolute value entry of an R8MAT.
Discussion:
An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
in column-major order.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
07 September 2012
Author:
John Burkardt
Parameters:
Input, int M, the number of rows in A.
Input, int N, the number of columns in A.
Input, double A[M*N], the M by N matrix.
Output, double R8MAT_AMAX, the maximum absolute value entry of A.
*/
{
double value = r8_abs(a[0 + 0 * m]);
for (int j = 0; j < n; j++) {
for (int i = 0; i < m; i++) {
NOLESS(value, r8_abs(a[i + j * m]));
}
}
return value;
}
void r8mat_copy(double a2[], int m, int n, double a1[])
/******************************************************************************/
/**
Purpose:
R8MAT_COPY_NEW copies one R8MAT to a "new" R8MAT.
Discussion:
An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
in column-major order.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
26 July 2008
Author:
John Burkardt
Parameters:
Input, int M, N, the number of rows and columns.
Input, double A1[M*N], the matrix to be copied.
Output, double R8MAT_COPY_NEW[M*N], the copy of A1.
*/
{
for (int j = 0; j < n; j++) {
for (int i = 0; i < m; i++)
a2[i + j * m] = a1[i + j * m];
}
}
/******************************************************************************/
void daxpy(int n, double da, double dx[], int incx, double dy[], int incy)
/******************************************************************************/
/**
Purpose:
DAXPY computes constant times a vector plus a vector.
Discussion:
This routine uses unrolled loops for increments equal to one.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
30 March 2007
Author:
C version by John Burkardt
Reference:
Jack Dongarra, Cleve Moler, Jim Bunch, Pete Stewart,
LINPACK User's Guide,
SIAM, 1979.
Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
Basic Linear Algebra Subprograms for Fortran Usage,
Algorithm 539,
ACM Transactions on Mathematical Software,
Volume 5, Number 3, September 1979, pages 308-323.
Parameters:
Input, int N, the number of elements in DX and DY.
Input, double DA, the multiplier of DX.
Input, double DX[*], the first vector.
Input, int INCX, the increment between successive entries of DX.
Input/output, double DY[*], the second vector.
On output, DY[*] has been replaced by DY[*] + DA * DX[*].
Input, int INCY, the increment between successive entries of DY.
*/
{
if (n <= 0 || da == 0.0) return;
int i, ix, iy, m;
/**
Code for unequal increments or equal increments
not equal to 1.
*/
if (incx != 1 || incy != 1) {
if (0 <= incx)
ix = 0;
else
ix = (- n + 1) * incx;
if (0 <= incy)
iy = 0;
else
iy = (- n + 1) * incy;
for (i = 0; i < n; i++) {
dy[iy] = dy[iy] + da * dx[ix];
ix = ix + incx;
iy = iy + incy;
}
}
/**
Code for both increments equal to 1.
*/
else {
m = n % 4;
for (i = 0; i < m; i++)
dy[i] = dy[i] + da * dx[i];
for (i = m; i < n; i = i + 4) {
dy[i ] = dy[i ] + da * dx[i ];
dy[i + 1] = dy[i + 1] + da * dx[i + 1];
dy[i + 2] = dy[i + 2] + da * dx[i + 2];
dy[i + 3] = dy[i + 3] + da * dx[i + 3];
}
}
}
/******************************************************************************/
double ddot(int n, double dx[], int incx, double dy[], int incy)
/******************************************************************************/
/**
Purpose:
DDOT forms the dot product of two vectors.
Discussion:
This routine uses unrolled loops for increments equal to one.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
30 March 2007
Author:
C version by John Burkardt
Reference:
Jack Dongarra, Cleve Moler, Jim Bunch, Pete Stewart,
LINPACK User's Guide,
SIAM, 1979.
Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
Basic Linear Algebra Subprograms for Fortran Usage,
Algorithm 539,
ACM Transactions on Mathematical Software,
Volume 5, Number 3, September 1979, pages 308-323.
Parameters:
Input, int N, the number of entries in the vectors.
Input, double DX[*], the first vector.
Input, int INCX, the increment between successive entries in DX.
Input, double DY[*], the second vector.
Input, int INCY, the increment between successive entries in DY.
Output, double DDOT, the sum of the product of the corresponding
entries of DX and DY.
*/
{
if (n <= 0) return 0.0;
int i, m;
double dtemp = 0.0;
/**
Code for unequal increments or equal increments
not equal to 1.
*/
if (incx != 1 || incy != 1) {
int ix = (incx >= 0) ? 0 : (-n + 1) * incx,
iy = (incy >= 0) ? 0 : (-n + 1) * incy;
for (i = 0; i < n; i++) {
dtemp += dx[ix] * dy[iy];
ix = ix + incx;
iy = iy + incy;
}
}
/**
Code for both increments equal to 1.
*/
else {
m = n % 5;
for (i = 0; i < m; i++)
dtemp += dx[i] * dy[i];
for (i = m; i < n; i = i + 5) {
dtemp += dx[i] * dy[i]
+ dx[i + 1] * dy[i + 1]
+ dx[i + 2] * dy[i + 2]
+ dx[i + 3] * dy[i + 3]
+ dx[i + 4] * dy[i + 4];
}
}
return dtemp;
}
/******************************************************************************/
double dnrm2(int n, double x[], int incx)
/******************************************************************************/
/**
Purpose:
DNRM2 returns the euclidean norm of a vector.
Discussion:
DNRM2 ( X ) = sqrt ( X' * X )
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
30 March 2007
Author:
C version by John Burkardt
Reference:
Jack Dongarra, Cleve Moler, Jim Bunch, Pete Stewart,
LINPACK User's Guide,
SIAM, 1979.
Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
Basic Linear Algebra Subprograms for Fortran Usage,
Algorithm 539,
ACM Transactions on Mathematical Software,
Volume 5, Number 3, September 1979, pages 308-323.
Parameters:
Input, int N, the number of entries in the vector.
Input, double X[*], the vector whose norm is to be computed.
Input, int INCX, the increment between successive entries of X.
Output, double DNRM2, the Euclidean norm of X.
*/
{
double norm;
if (n < 1 || incx < 1)
norm = 0.0;
else if (n == 1)
norm = r8_abs(x[0]);
else {
double scale = 0.0, ssq = 1.0;
int ix = 0;
for (int i = 0; i < n; i++) {
if (x[ix] != 0.0) {
double absxi = r8_abs(x[ix]);
if (scale < absxi) {
ssq = 1.0 + ssq * (scale / absxi) * (scale / absxi);
scale = absxi;
}
else
ssq = ssq + (absxi / scale) * (absxi / scale);
}
ix += incx;
}
norm = scale * sqrt(ssq);
}
return norm;
}
/******************************************************************************/
void dqrank(double a[], int lda, int m, int n, double tol, int* kr,
int jpvt[], double qraux[])
/******************************************************************************/
/**
Purpose:
DQRANK computes the QR factorization of a rectangular matrix.
Discussion:
This routine is used in conjunction with DQRLSS to solve
overdetermined, underdetermined and singular linear systems
in a least squares sense.
DQRANK uses the LINPACK subroutine DQRDC to compute the QR
factorization, with column pivoting, of an M by N matrix A.
The numerical rank is determined using the tolerance TOL.
Note that on output, ABS ( A(1,1) ) / ABS ( A(KR,KR) ) is an estimate
of the condition number of the matrix of independent columns,
and of R. This estimate will be <= 1/TOL.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
21 April 2012
Author:
C version by John Burkardt.
Reference:
Jack Dongarra, Cleve Moler, Jim Bunch, Pete Stewart,
LINPACK User's Guide,
SIAM, 1979,
ISBN13: 978-0-898711-72-1,
LC: QA214.L56.
Parameters:
Input/output, double A[LDA*N]. On input, the matrix whose
decomposition is to be computed. On output, the information from DQRDC.
The triangular matrix R of the QR factorization is contained in the
upper triangle and information needed to recover the orthogonal
matrix Q is stored below the diagonal in A and in the vector QRAUX.
Input, int LDA, the leading dimension of A, which must
be at least M.
Input, int M, the number of rows of A.
Input, int N, the number of columns of A.
Input, double TOL, a relative tolerance used to determine the
numerical rank. The problem should be scaled so that all the elements
of A have roughly the same absolute accuracy, EPS. Then a reasonable
value for TOL is roughly EPS divided by the magnitude of the largest
element.
Output, int *KR, the numerical rank.
Output, int JPVT[N], the pivot information from DQRDC.
Columns JPVT(1), ..., JPVT(KR) of the original matrix are linearly
independent to within the tolerance TOL and the remaining columns
are linearly dependent.
Output, double QRAUX[N], will contain extra information defining
the QR factorization.
*/
{
double work[n];
for (int i = 0; i < n; i++)
jpvt[i] = 0;
int job = 1;
dqrdc(a, lda, m, n, qraux, jpvt, work, job);
*kr = 0;
int k = i4_min(m, n);
for (int j = 0; j < k; j++) {
if (r8_abs(a[j + j * lda]) <= tol * r8_abs(a[0 + 0 * lda]))
return;
*kr = j + 1;
}
}
/******************************************************************************/
void dqrdc(double a[], int lda, int n, int p, double qraux[], int jpvt[],
double work[], int job)
/******************************************************************************/
/**
Purpose:
DQRDC computes the QR factorization of a real rectangular matrix.
Discussion:
DQRDC uses Householder transformations.
Column pivoting based on the 2-norms of the reduced columns may be
performed at the user's option.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
07 June 2005
Author:
C version by John Burkardt.
Reference:
Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
LINPACK User's Guide,
SIAM, (Society for Industrial and Applied Mathematics),
3600 University City Science Center,
Philadelphia, PA, 19104-2688.
ISBN 0-89871-172-X
Parameters:
Input/output, double A(LDA,P). On input, the N by P matrix
whose decomposition is to be computed. On output, A contains in
its upper triangle the upper triangular matrix R of the QR
factorization. Below its diagonal A contains information from
which the orthogonal part of the decomposition can be recovered.
Note that if pivoting has been requested, the decomposition is not that
of the original matrix A but that of A with its columns permuted
as described by JPVT.
Input, int LDA, the leading dimension of the array A. LDA must
be at least N.
Input, int N, the number of rows of the matrix A.
Input, int P, the number of columns of the matrix A.
Output, double QRAUX[P], contains further information required
to recover the orthogonal part of the decomposition.
Input/output, integer JPVT[P]. On input, JPVT contains integers that
control the selection of the pivot columns. The K-th column A(*,K) of A
is placed in one of three classes according to the value of JPVT(K).
> 0, then A(K) is an initial column.
= 0, then A(K) is a free column.
< 0, then A(K) is a final column.
Before the decomposition is computed, initial columns are moved to
the beginning of the array A and final columns to the end. Both
initial and final columns are frozen in place during the computation
and only free columns are moved. At the K-th stage of the
reduction, if A(*,K) is occupied by a free column it is interchanged
with the free column of largest reduced norm. JPVT is not referenced
if JOB == 0. On output, JPVT(K) contains the index of the column of the
original matrix that has been interchanged into the K-th column, if
pivoting was requested.
Workspace, double WORK[P]. WORK is not referenced if JOB == 0.
Input, int JOB, initiates column pivoting.
0, no pivoting is done.
nonzero, pivoting is done.
*/
{
int jp;
int j;
int lup;
int maxj;
double maxnrm, nrmxl, t, tt;
int pl = 1, pu = 0;
/**
If pivoting is requested, rearrange the columns.
*/
if (job != 0) {
for (j = 1; j <= p; j++) {
int swapj = (0 < jpvt[j - 1]);
jpvt[j - 1] = (jpvt[j - 1] < 0) ? -j : j;
if (swapj) {
if (j != pl)
dswap(n, a + 0 + (pl - 1)*lda, 1, a + 0 + (j - 1), 1);
jpvt[j - 1] = jpvt[pl - 1];
jpvt[pl - 1] = j;
pl++;
}
}
pu = p;
for (j = p; 1 <= j; j--) {
if (jpvt[j - 1] < 0) {
jpvt[j - 1] = -jpvt[j - 1];
if (j != pu) {
dswap(n, a + 0 + (pu - 1)*lda, 1, a + 0 + (j - 1)*lda, 1);
jp = jpvt[pu - 1];
jpvt[pu - 1] = jpvt[j - 1];
jpvt[j - 1] = jp;
}
pu = pu - 1;
}
}
}
/**
Compute the norms of the free columns.
*/
for (j = pl; j <= pu; j++)
qraux[j - 1] = dnrm2(n, a + 0 + (j - 1) * lda, 1);
for (j = pl; j <= pu; j++)
work[j - 1] = qraux[j - 1];
/**
Perform the Householder reduction of A.
*/
lup = i4_min(n, p);
for (int l = 1; l <= lup; l++) {
/**
Bring the column of largest norm into the pivot position.
*/
if (pl <= l && l < pu) {
maxnrm = 0.0;
maxj = l;
for (j = l; j <= pu; j++) {
if (maxnrm < qraux[j - 1]) {
maxnrm = qraux[j - 1];
maxj = j;
}
}
if (maxj != l) {
dswap(n, a + 0 + (l - 1)*lda, 1, a + 0 + (maxj - 1)*lda, 1);
qraux[maxj - 1] = qraux[l - 1];
work[maxj - 1] = work[l - 1];
jp = jpvt[maxj - 1];
jpvt[maxj - 1] = jpvt[l - 1];
jpvt[l - 1] = jp;
}
}
/**
Compute the Householder transformation for column L.
*/
qraux[l - 1] = 0.0;
if (l != n) {
nrmxl = dnrm2(n - l + 1, a + l - 1 + (l - 1) * lda, 1);
if (nrmxl != 0.0) {
if (a[l - 1 + (l - 1)*lda] != 0.0)
nrmxl = nrmxl * r8_sign(a[l - 1 + (l - 1) * lda]);
dscal(n - l + 1, 1.0 / nrmxl, a + l - 1 + (l - 1)*lda, 1);
a[l - 1 + (l - 1)*lda] = 1.0 + a[l - 1 + (l - 1) * lda];
/**
Apply the transformation to the remaining columns, updating the norms.
*/
for (j = l + 1; j <= p; j++) {
t = -ddot(n - l + 1, a + l - 1 + (l - 1) * lda, 1, a + l - 1 + (j - 1) * lda, 1)
/ a[l - 1 + (l - 1) * lda];
daxpy(n - l + 1, t, a + l - 1 + (l - 1)*lda, 1, a + l - 1 + (j - 1)*lda, 1);
if (pl <= j && j <= pu) {
if (qraux[j - 1] != 0.0) {
tt = 1.0 - pow(r8_abs(a[l - 1 + (j - 1) * lda]) / qraux[j - 1], 2);
tt = r8_max(tt, 0.0);
t = tt;
tt = 1.0 + 0.05 * tt * pow(qraux[j - 1] / work[j - 1], 2);
if (tt != 1.0)
qraux[j - 1] = qraux[j - 1] * sqrt(t);
else {
qraux[j - 1] = dnrm2(n - l, a + l + (j - 1) * lda, 1);
work[j - 1] = qraux[j - 1];
}
}
}
}
/**
Save the transformation.
*/
qraux[l - 1] = a[l - 1 + (l - 1) * lda];
a[l - 1 + (l - 1)*lda] = -nrmxl;
}
}
}
}
/******************************************************************************/
int dqrls(double a[], int lda, int m, int n, double tol, int* kr, double b[],
double x[], double rsd[], int jpvt[], double qraux[], int itask)
/******************************************************************************/
/**
Purpose:
DQRLS factors and solves a linear system in the least squares sense.
Discussion:
The linear system may be overdetermined, underdetermined or singular.
The solution is obtained using a QR factorization of the
coefficient matrix.
DQRLS can be efficiently used to solve several least squares
problems with the same matrix A. The first system is solved
with ITASK = 1. The subsequent systems are solved with
ITASK = 2, to avoid the recomputation of the matrix factors.
The parameters KR, JPVT, and QRAUX must not be modified
between calls to DQRLS.
DQRLS is used to solve in a least squares sense
overdetermined, underdetermined and singular linear systems.
The system is A*X approximates B where A is M by N.
B is a given M-vector, and X is the N-vector to be computed.
A solution X is found which minimimzes the sum of squares (2-norm)
of the residual, A*X - B.
The numerical rank of A is determined using the tolerance TOL.
DQRLS uses the LINPACK subroutine DQRDC to compute the QR
factorization, with column pivoting, of an M by N matrix A.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
10 September 2012
Author:
C version by John Burkardt.
Reference:
David Kahaner, Cleve Moler, Steven Nash,
Numerical Methods and Software,
Prentice Hall, 1989,
ISBN: 0-13-627258-4,
LC: TA345.K34.
Parameters:
Input/output, double A[LDA*N], an M by N matrix.
On input, the matrix whose decomposition is to be computed.
In a least squares data fitting problem, A(I,J) is the
value of the J-th basis (model) function at the I-th data point.
On output, A contains the output from DQRDC. The triangular matrix R
of the QR factorization is contained in the upper triangle and
information needed to recover the orthogonal matrix Q is stored
below the diagonal in A and in the vector QRAUX.
Input, int LDA, the leading dimension of A.
Input, int M, the number of rows of A.
Input, int N, the number of columns of A.
Input, double TOL, a relative tolerance used to determine the
numerical rank. The problem should be scaled so that all the elements
of A have roughly the same absolute accuracy EPS. Then a reasonable
value for TOL is roughly EPS divided by the magnitude of the largest
element.
Output, int *KR, the numerical rank.
Input, double B[M], the right hand side of the linear system.
Output, double X[N], a least squares solution to the linear
system.
Output, double RSD[M], the residual, B - A*X. RSD may
overwrite B.
Workspace, int JPVT[N], required if ITASK = 1.
Columns JPVT(1), ..., JPVT(KR) of the original matrix are linearly
independent to within the tolerance TOL and the remaining columns
are linearly dependent. ABS ( A(1,1) ) / ABS ( A(KR,KR) ) is an estimate
of the condition number of the matrix of independent columns,
and of R. This estimate will be <= 1/TOL.
Workspace, double QRAUX[N], required if ITASK = 1.
Input, int ITASK.
1, DQRLS factors the matrix A and solves the least squares problem.
2, DQRLS assumes that the matrix A was factored with an earlier
call to DQRLS, and only solves the least squares problem.
Output, int DQRLS, error code.
0: no error
-1: LDA < M (fatal error)
-2: N < 1 (fatal error)
-3: ITASK < 1 (fatal error)
*/
{
int ind;
if (lda < m) {
/*fprintf ( stderr, "\n" );
fprintf ( stderr, "DQRLS - Fatal error!\n" );
fprintf ( stderr, " LDA < M.\n" );*/
ind = -1;
return ind;
}
if (n <= 0) {
/*fprintf ( stderr, "\n" );
fprintf ( stderr, "DQRLS - Fatal error!\n" );
fprintf ( stderr, " N <= 0.\n" );*/
ind = -2;
return ind;
}
if (itask < 1) {
/*fprintf ( stderr, "\n" );
fprintf ( stderr, "DQRLS - Fatal error!\n" );
fprintf ( stderr, " ITASK < 1.\n" );*/
ind = -3;
return ind;
}
ind = 0;
/**
Factor the matrix.
*/
if (itask == 1)
dqrank(a, lda, m, n, tol, kr, jpvt, qraux);
/**
Solve the least-squares problem.
*/
dqrlss(a, lda, m, n, *kr, b, x, rsd, jpvt, qraux);
return ind;
}
/******************************************************************************/
void dqrlss(double a[], int lda, int m, int n, int kr, double b[], double x[],
double rsd[], int jpvt[], double qraux[])
/******************************************************************************/
/**
Purpose:
DQRLSS solves a linear system in a least squares sense.
Discussion:
DQRLSS must be preceded by a call to DQRANK.
The system is to be solved is
A * X = B
where
A is an M by N matrix with rank KR, as determined by DQRANK,
B is a given M-vector,
X is the N-vector to be computed.
A solution X, with at most KR nonzero components, is found which
minimizes the 2-norm of the residual (A*X-B).
Once the matrix A has been formed, DQRANK should be
called once to decompose it. Then, for each right hand
side B, DQRLSS should be called once to obtain the
solution and residual.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
10 September 2012