#! /usr/bin/env python # def qr_solve ( m, n, a, b ): #*****************************************************************************80 # ## QR_SOLVE solves a linear system in the least squares sense. # # Discussion: # # If the matrix A has full column rank, then the solution X should be the # unique vector that minimizes the Euclidean norm of the residual. # # If the matrix A does not have full column rank, then the solution is # not unique; the vector X will minimize the residual norm, but so will # various other vectors. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 27 August 2016 # # Author: # # 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, integer M, the number of rows of A. # # Input, integer N, the number of columns of A. # # Input, real A(M,N), the matrix. # # Input, real B(M), the right hand side. # # Output, real X(N), the least squares solution. # from dqrank import dqrank from dqrlss import dqrlss lda = m eps = 2.220446049250313E-016 tol = 0.0 for i in range ( 0, m ): for j in range ( 0, n ): tol = max ( tol, abs ( a[i,j] ) ) tol = eps / tol # # Factor the matrix. # kr, jpvt, qraux, a = dqrank ( a, lda, m, n, tol ) # # Solve the least-squares problem. # x, r = dqrlss ( a, lda, m, n, kr, b, jpvt, qraux ) return x def qr_solve_test ( ): #*****************************************************************************80 # ## QR_SOLVE_TEST tests QR_SOLVE. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 26 August 2016 # # Author: # # John Burkardt # import numpy as np import platform from r8vec_norm import r8vec_norm from test_ls import p00_a from test_ls import p00_b from test_ls import p00_m from test_ls import p00_n from test_ls import p00_prob_num from test_ls import p00_x print ( '' ) print ( 'QR_SOLVE_TEST' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' QR_SOLVE is a function with a simple interface which' ) print ( ' solves a linear system A*x = b in the least squares sense.' ) print ( ' Compare a tabulated solution X1 to the QR_SOLVE result X2.' ) prob_num = p00_prob_num ( ) print ( '' ) print ( ' Number of problems = %d' % ( prob_num ) ) print ( '' ) print ( ' Index M N ||B|| ||X1 - X2||' ), print ( ' ||X1|| ||X2|| ||R1|| ||R2||' ) print ( '' ) for prob in range ( 1, prob_num + 1 ): # # Get problem size. # m = p00_m ( prob ) n = p00_n ( prob ) # # Retrieve problem data. # a = p00_a ( prob, m, n ) b = p00_b ( prob, m ) x1 = p00_x ( prob, n ) b_norm = r8vec_norm ( m, b ) x1_norm = r8vec_norm ( n, x1 ) r1 = np.dot ( a, x1 ) - b r1_norm = r8vec_norm ( m, r1 ) # # Use QR_SOLVE on the problem. # Since we want to compute residuals ourselves, we need # to keep the originals of A and B, so we make copies # to send to QR_SOLVE. # a_qr = a.copy ( ) b_qr = b.copy ( ) x2 = qr_solve ( m, n, a_qr, b_qr ) x2_norm = r8vec_norm ( n, x2 ) r2 = np.dot ( a, x2 ) - b r2_norm = r8vec_norm ( m, r2 ) # # Compare tabulated and computed solutions. # x_diff_norm = r8vec_norm ( n, x1 - x2 ) # # Report results for this problem. # print ( ' %5d %4d %4d' % ( prob, m, n ) ), print ( ' %12.4g %12.4g' % ( b_norm, x_diff_norm ) ), print ( ' %12.4g %12.4g' % ( x1_norm, x2_norm ) ), print ( ' %12.4g %12.4g' % ( r1_norm, r2_norm ) ) # # Terminate. # print ( '' ) print ( 'QR_SOLVE_TEST' ) print ( ' Normal end of execution.' ) return if ( __name__ == '__main__' ): from timestamp import timestamp timestamp ( ) qr_solve_test ( ) timestamp ( )