lsqr.txt

比较经典的求解线性方程的方法 原理是C.C. Paige and M.A. Sauders等你提出的

     4  8.372079751E+00  1.127717708E+02  1.09E-01  2.11E-01  1.51E+00  5.07E+00  1.5E-01
     5  1.186968037E+01  8.222083684E+01  7.93E-02  1.65E-01  1.61E+00  6.84E+00  1.1E-01
     6  1.189447236E+01  6.123052191E+01  5.91E-02  1.31E-01  1.68E+00  8.87E+00  8.5E-02
     7  9.555842886E+00  4.606250067E+01  4.44E-02  1.04E-01  1.73E+00  1.12E+01  6.6E-02
     8  5.773292099E+00  3.472664457E+01  3.35E-02  8.33E-02  1.76E+00  1.40E+01  5.1E-02
     9  1.300148876E+00  2.607350378E+01  2.52E-02  6.60E-02  1.79E+00  1.75E+01  4.0E-02
    10 -3.232558669E+00  1.939324792E+01  1.87E-02  5.18E-02  1.80E+00  2.18E+01  3.1E-02
    20 -1.047231653E+01  2.009566353E+00  1.94E-03  7.17E-03  2.40E+00  1.46E+02  4.3E-03
    30 -2.134966879E+00  3.220999881E-01  3.11E-04  6.22E-03  2.93E+00  6.24E+02  9.1E-04
    40  5.387315818E-01  1.083027649E-01  1.05E-04  1.77E-02  3.36E+00  1.87E+03  3.3E-04
    50  1.491068356E+00  2.189241802E-02  2.11E-05  4.13E-02  3.76E+00  5.93E+03  8.7E-05
    60  1.329628283E+00  8.515005934E-03  8.22E-06  6.78E-03  4.12E+00  1.54E+04  3.5E-05
    70  1.107467931E+00  1.492425153E-03  1.44E-06  5.33E-05  4.38E+00  2.24E+04  1.3E-05
    80  8.599660800E-01  1.163792248E-03  1.12E-06  6.21E-02  4.70E+00  7.58E+04  6.3E-06
    90  4.763538974E-01  4.534520291E-05  4.38E-08  4.87E-03  5.02E+00  1.25E+05  1.0E-06
   100  4.760759205E-01  3.534402254E-05  3.41E-08  4.02E-06  5.27E+00  1.32E+05  8.8E-07
   110  4.575445334E-01  3.445936533E-05  3.33E-08  4.77E-03  5.52E+00  4.99E+05  4.6E-07
   120  2.149322613E-01  1.953702888E-05  1.89E-08  2.44E-02  5.78E+00  1.89E+06  1.8E-07
   130  1.005061144E-01  1.312014930E-06  1.27E-09  4.41E-02  6.01E+00  2.35E+06  4.3E-08
   140  1.000000177E-01  1.827420739E-07  1.76E-10  7.38E-05  6.26E+00  2.45E+06  1.6E-08
   150  9.999999967E-02  1.827111280E-07  1.76E-10  4.97E-05  6.43E+00  2.52E+06  1.6E-08
   160  9.999999966E-02  1.827111116E-07  1.76E-10  1.25E-07  6.66E+00  2.61E+06  1.6E-08
   170  9.999999999E-02  1.827111108E-07  1.76E-10  1.19E-08  6.84E+00  2.68E+06  1.6E-08
   180  1.000000000E-01  1.827111108E-07  1.76E-10  8.78E-12  7.04E+00  2.76E+06  1.6E-08
   190  1.000000000E-01  1.827111108E-07  1.76E-10  8.98E-10  7.25E+00  2.84E+06  1.6E-08
   200  1.000000000E-01  1.827111108E-07  1.76E-10  1.94E-09  7.46E+00  2.92E+06  1.6E-08
   210  1.000000000E-01  1.827111108E-07  1.76E-10  1.82E-12  7.64E+00  3.00E+06  1.6E-08
   216  1.000000000E-01  1.827111108E-07  1.76E-10  2.97E-15  7.74E+00  3.04E+06  1.6E-08
   217  1.000000000E-01  1.827111108E-07  1.76E-10  1.11E-15  7.75E+00  3.04E+06  1.6E-08
   218  1.000000000E-01  1.827111108E-07  1.76E-10  2.94E-16  7.79E+00  3.05E+06  1.6E-08

 Exit  LSQR.       istop  = 3               itn    =     218
 Exit  LSQR.       Anorm  = 7.78844E+00     Acond  = 3.05475E+06
 Exit  LSQR.       bnorm  = 1.03632E+03     xnorm  = 1.82711E+03
 Exit  LSQR.       rnorm  = 1.82711E-07     Arnorm = 4.19012E-22
 Exit  LSQR.       max dx = 1.1E+03 occurred at itn        1
 Exit  LSQR.              = 5.8E-01*xnorm
 Exit  LSQR.       A damped least-squares solution was found, given atol


 Enter xcheck.     Does x solve Ax = b, etc?
    damp            = 1.000E-10
    norm(x)         = 1.827E+03
    norm(r)         = 7.90044355E-13 = rho1
    norm(A'r)       = 6.403E-13      = sigma1

    norm(s)         = 7.900E-03
    norm(x,s)       = 1.827E+03
    norm(rbar)      = 1.82711111E-07 = rho2
    norm(Abar'rbar) = 6.403E-13      = sigma2

    inform          = 1
    tol             = 1.490E-08
    test1           = 5.175E-17 (Ax = b)
    test2           = 1.041E-01 (least-squares)
    test3           = 4.500E-07 (damped least-squares)


 Solution  x:
     1  0.100000         2  0.200000         3  0.300000         4  0.400000    
     5  0.500000         6  0.600000         7  0.700000         8  0.800000    


 LSQR  appears to be successful.
 Relative error in  x  =  1.52E-12


 --------------------------------------------------------------------
 Least-Squares Test Problem      P( 1000 1000   40    5    1.00E-11 )
 Condition no. =  9.7656E+06     Residual function =  1.359023811E-14
 --------------------------------------------------------------------


 Enter Acheck.     Test of Aprod for LSQR and CRAIG
 Aprod seems OK.   Relative error =   4.7E-16


 Enter LSQR.       Least-squares solution of  Ax = b
 The matrix  A  has   1000 rows   and   1000 columns
 damp   =  1.00000000000000E-11   wantse =         F
 atol   =  3.18E-16               conlim =  9.77E+09
 btol   =  3.18E-16               itnlim =      8200


   Itn       x(1)           Function     Compatible   LS     Norm Abar Cond Abar alfa_opt
     0  0.000000000E+00  9.710346493E+02  1.00E+00  8.45E-04
     1 -2.412106256E+01  4.104497706E+02  4.23E-01  6.36E-01  9.06E-01  1.00E+00  6.2E-01
     2 -1.824596476E+01  2.375711335E+02  2.45E-01  3.77E-01  1.18E+00  2.18E+00  3.3E-01
     3 -7.118201010E+00  1.553550200E+02  1.60E-01  2.61E-01  1.36E+00  3.57E+00  2.1E-01
     4  2.622411370E+00  1.077493654E+02  1.11E-01  1.92E-01  1.47E+00  5.19E+00  1.5E-01
     5  9.349109039E+00  7.695260886E+01  7.92E-02  1.45E-01  1.55E+00  7.12E+00  1.1E-01
     6  1.286861549E+01  5.564981711E+01  5.73E-02  1.11E-01  1.60E+00  9.44E+00  7.8E-02
     7  1.344322975E+01  4.031108974E+01  4.15E-02  8.47E-02  1.63E+00  1.23E+01  5.8E-02
     8  1.153767000E+01  2.902297219E+01  2.99E-02  6.43E-02  1.65E+00  1.60E+01  4.3E-02
     9  7.760937429E+00  2.064415371E+01  2.13E-02  4.82E-02  1.66E+00  2.08E+01  3.2E-02
    10  2.839814774E+00  1.443334690E+01  1.49E-02  3.55E-02  1.67E+00  2.74E+01  2.3E-02
    20 -1.337282972E+01  1.551066928E+00  1.60E-03  4.62E-03  2.30E+00  2.02E+02  3.2E-03
    30 -6.107927383E+00  2.143050643E-01  2.21E-04  1.34E-02  2.83E+00  1.06E+03  5.6E-04
    40 -2.843352165E+00  8.811598200E-02  9.07E-05  7.21E-04  3.31E+00  2.31E+03  2.6E-04
    50  8.763593882E-01  1.628310573E-02  1.68E-05  1.03E-01  3.69E+00  1.15E+04  5.4E-05
    60  1.204414332E+00  8.290757129E-03  8.54E-06  5.15E-04  4.02E+00  1.43E+04  3.6E-05
    70  1.561179786E+00  1.488426091E-03  1.53E-06  2.14E-04  4.34E+00  4.41E+04  9.0E-06
    80  1.557919355E+00  1.482552221E-03  1.53E-06  2.43E-03  4.61E+00  4.95E+04  8.7E-06
    90  1.126165007E+00  1.474203082E-04  1.52E-07  4.84E-04  4.89E+00  2.03E+05  1.4E-06
   100  1.124359697E+00  1.225147551E-04  1.26E-07  3.32E-06  5.19E+00  2.15E+05  1.3E-06
   110  1.124341926E+00  1.225118504E-04  1.26E-07  1.35E-04  5.49E+00  2.28E+05  1.3E-06
   120  4.885035761E-01  1.591113876E-05  1.64E-08  3.42E-02  5.70E+00  1.74E+06  1.7E-07
   130  4.772550558E-01  1.524660556E-06  1.57E-09  1.69E-02  5.91E+00  1.82E+06  5.2E-08
   140  4.772408735E-01  1.418278540E-06  1.46E-09  5.53E-06  6.12E+00  1.89E+06  5.0E-08
   150  4.772413763E-01  1.418235947E-06  1.46E-09  3.73E-07  6.33E+00  1.95E+06  5.0E-08
   160  4.772413742E-01  1.418235611E-06  1.46E-09  2.56E-06  6.54E+00  2.02E+06  5.0E-08
   170  4.754554192E-01  1.414871820E-06  1.46E-09  9.97E-04  6.80E+00  5.03E+06  3.2E-08
   180  4.634786689E-01  1.392100977E-06  1.43E-09  6.79E-04  6.99E+00  1.32E+07  2.0E-08
   190  4.453658616E-01  1.356977886E-06  1.40E-09  1.48E-03  7.17E+00  2.05E+07  1.6E-08
   200  1.000027155E-01  1.865230402E-08  1.92E-11  5.75E-04  7.32E+00  7.15E+07  1.0E-09
   210  9.999998038E-02  1.827228994E-08  1.88E-11  1.17E-05  7.50E+00  7.33E+07  1.0E-09
   220  9.999997662E-02  1.827186009E-08  1.88E-11  5.85E-07  7.67E+00  7.50E+07  1.0E-09
   230  1.000000009E-01  1.827111513E-08  1.88E-11  3.17E-05  7.90E+00  7.72E+07  1.0E-09
   240  1.000000010E-01  1.827111209E-08  1.88E-11  1.79E-07  8.07E+00  7.88E+07  1.0E-09
   250  1.000000012E-01  1.827111134E-08  1.88E-11  3.06E-08  8.22E+00  8.03E+07  1.0E-09
   260  1.000000013E-01  1.827111108E-08  1.88E-11  2.70E-11  8.38E+00  8.19E+07  1.0E-09
   270  1.000000013E-01  1.827111108E-08  1.88E-11  9.09E-11  8.53E+00  8.33E+07  1.0E-09
   280  1.000000014E-01  1.827111108E-08  1.88E-11  1.52E-08  8.66E+00  8.47E+07  1.0E-09
   290  1.000000014E-01  1.827111108E-08  1.88E-11  1.73E-09  8.82E+00  8.62E+07  1.0E-09
   300  1.000000014E-01  1.827111108E-08  1.88E-11  1.27E-10  9.00E+00  8.80E+07  1.0E-09
   310  1.000000014E-01  1.827111108E-08  1.88E-11  9.47E-11  9.15E+00  8.94E+07  1.0E-09
   320  1.000000014E-01  1.827111108E-08  1.88E-11  4.76E-14  9.30E+00  9.09E+07  1.0E-09
   322  1.000000014E-01  1.827111108E-08  1.88E-11  1.30E-15  9.31E+00  9.10E+07  1.0E-09
   324  1.000000014E-01  1.827111108E-08  1.88E-11  5.30E-17  9.35E+00  9.14E+07  1.0E-09

 Exit  LSQR.       istop  = 3               itn    =     324
 Exit  LSQR.       Anorm  = 9.34839E+00     Acond  = 9.13837E+07
 Exit  LSQR.       bnorm  = 9.71035E+02     xnorm  = 1.82711E+03
 Exit  LSQR.       rnorm  = 1.82711E-08     Arnorm = 9.05576E-24
 Exit  LSQR.       max dx = 9.7E+02 occurred at itn        1
 Exit  LSQR.              = 5.3E-01*xnorm
 Exit  LSQR.       A damped least-squares solution was found, given atol


 Enter xcheck.     Does x solve Ax = b, etc?
    damp            = 1.000E-11
    norm(x)         = 1.827E+03
    norm(r)         = 9.31998493E-13 = rho1
    norm(A'r)       = 7.480E-13      = sigma1

    norm(s)         = 9.320E-02
    norm(x,s)       = 1.827E+03
    norm(rbar)      = 1.82711111E-08 = rho2
    norm(Abar'rbar) = 7.480E-13      = sigma2

    inform          = 1
    tol             = 1.490E-08
    test1           = 5.163E-17 (Ax = b)
    test2           = 8.586E-02 (least-squares)
    test3           = 4.380E-06 (damped least-squares)


 Solution  x:
     1  0.100000         2  0.200000         3  0.300000         4  0.400000    
     5  0.500000         6  0.600000         7  0.700000         8  0.800000    


 LSQR  appears to be successful.
 Relative error in  x  =  7.27E-11


 --------------------------------------------------------------------
 Least-Squares Test Problem      P( 1000 1000   40    6    1.00E-12 )
 Condition no. =  2.4414E+08     Residual function =  3.387612114E-15
 --------------------------------------------------------------------


 Enter Acheck.     Test of Aprod for LSQR and CRAIG
 Aprod seems OK.   Relative error =   4.1E-16


 Enter LSQR.       Least-squares solution of  Ax = b
 The matrix  A  has   1000 rows   and   1000 columns
 damp   =  1.00000000000000E-12   wantse =         F
 atol   =  3.18E-16               conlim =  2.44E+11
 btol   =  3.18E-16               itnlim =      8200


   Itn       x(1)           Function     Compatible   LS     Norm Abar Cond Abar alfa_opt
     0  0.000000000E+00  9.205361605E+02  1.00E+00  8.98E-04
     1 -2.443055276E+01  3.938615198E+02  4.28E-01  6.20E-01  9.14E-01  1.00E+00  6.3E-01
     2 -2.224414133E+01  2.277481391E+02  2.47E-01  3.61E-01  1.18E+00  2.19E+00  3.3E-01
     3 -1.321097804E+01  1.472531389E+02  1.60E-01  2.44E-01  1.34E+00  3.62E+00  2.1E-01
     4 -3.367416062E+00  1.000443384E+02  1.09E-01  1.74E-01  1.44E+00  5.35E+00  1.4E-01
     5  5.036316961E+00  6.938393962E+01  7.54E-02  1.27E-01  1.50E+00  7.50E+00  9.8E-02
     6  1.099397591E+01  4.832729386E+01  5.25E-02  9.29E-02  1.53E+00  1.02E+01  6.9E-02
     7  1.404999112E+01  3.345806131E+01  3.63E-02  6.76E-02  1.55E+00  1.39E+01  4.9E-02
     8  1.413848446E+01  2.285799571E+01  2.48E-02  4.86E-02  1.56E+00  1.88E+01  3.4E-02
     9  1.156137830E+01  1.532415806E+01  1.66E-02  3.43E-02  1.57E+00  2.59E+01  2.4E-02
    10  6.963704534E+00  1.003283262E+01  1.09E-02  2.37E-02  1.57E+00  3.61E+01  1.6E-02
    20 -1.367783570E+01  7.441206870E-01  8.08E-04  1.45E-01  2.19E+00  3.92E+02  1.5E-03
    30 -1.024542645E+01  1.591939739E-01  1.73E-04  8.04E-04  2.85E+00  1.60E+03  4.0E-04
    40 -6.485088123E+00  6.255582473E-02  6.80E-05  1.64E-02  3.21E+00  4.03E+03  1.7E-04
    50 -3.048693404E+00  2.204365586E-02  2.39E-05  1.62E-03  3.57E+00  9.93E+03  6.6E-05
    60 -4.549421868E-01  6.567702172E-03  7.13E-06  4.38E-05  3.89E+00  2.24E+04  2.5E-05
    70  1.171426557E+00  1.396948576E-03  1.52E-06  4.21E-05  4.23E+00  7.06E+04  6.8E-06
    80  1.171443702E+00  1.396881461E-03  1.52E-06  1.02E-04  4.45E+00  7.43E+04  6.8E-06
    90  1.568854229E+00  2.387824150E-04  2.59E-07  1.28E-02  4.80E+00  2.95E+05  1.5E-06
   100  1.574464644E+00  1.837749096E-04  2.00E-07  2.58E-06  5.06E+00  3.14E+05  1.3E-06
   110  1.571847882E+00  1.832715931E-04  1.99E-07  1.30E-03  5.29E+00  3.53E+05  1.2E-06
   120  1.131334993E+00  1.006954920E-05  1.09E-08  5.29E-03  5.52E+00  1.88E+06  1.3E-07
   130  1.131297078E+00  9.922704281E-06  1.08E-08  8.57E-04  5.73E+00  1.95E+06  1.3E-07
   140  1.131300719E+00  9.905113401E-06  1.08E-08  2.42E-06  6.00E+00  2.04E+06  1.3E-07
   150  1.131244601E+00  9.904643321E-06  1.08E-08  1.17E-05  6.21E+00  2.13E+06  1.3E-07
   160  1.131124500E+00  9.903649108E-06  1.08E-08  1.89E-05  6.38E+00  2.21E+06  1.3E-07
   170  4.777844616E-01  1.893260420E-07  2.06E-10  4.37E-04  6.56E+00  2.51E+07  5.2E-09
   180  4.777830043E-01  1.887415422E-07  2.05E-10  1.85E-04  6.74E+00  2.58E+07  5.2E-09
   190  4.777797851E-01  1.874107616E-07  2.04E-10  2.04E-05  6.96E+00  2.66E+07  5.2E-09
   200  4.775241976E-01  5.679709291E-08  6.17E-11  2.82E-06  7.15E+00  2.74E+07  2.8E-09
   210  4.775241976E-01  5.679708557E-08  6.17E-11  2.40E-06  7.32E+00  2.80E+07  2.8E-09
   220  4.775241972E-01  5.679707759E-08  6.17E-11  1.85E-07  7.48E+00  2.86E+07  2.8E-09
   230  4.775241139E-01  5.679662212E-08  6.17E-11  8.62E-07  7.63E+00  2.92E+07  2.8E-09
   240  4.775241079E-01  5.679661139E-08  6.17E-11  2.02E-06  7.84E+00  3.00E+07  2.8E-09
   250  4.775108727E-01  5.679561860E-08  6.17E-11  1.79E-05  8.00E+00  3.27E+07  2.8E-09
   260  4.775002093E-01  5.679481848E-08  6.17E-11  1.81E-06  8.14E+00  3.50E+07  2.7E-09
   270  4.208790381E-01  5.236700969E-08  5.69E-11  1.64E-03  8.29E+00  7.85E+08  5.5E-10
   280  1.158174720E-01  1.175348453E-08  1.28E-11  1.12E-04  8.46E+00  2.02E+09  1.6E-10
   290  1.158111819E-01  1.175120008E-08  1.28E-11  1.38E-05  8.62E+00  2.06E+09  1.6E-10
   300  1.157561020E-01  1.173121291E-08  1.27E-11  2.85E-03  8.76E+00  2.09E+09  1.6E-10
   310  1.105457946E-01  9.655295655E-09  1.05E-11  9.99E-05  8.92E+00  2.15E+09  1.5E-10
   320  1.000015981E-01  1.830608100E-09  1.99E-12  4.99E-05  9.05E+00  2.21E+09  6.4E-11
   330  1.000002170E-01  1.827394178E-09  1.99E-12  7.85E-06  9.18E+00  2.24E+09  6.4E-11
   340  1.000001058E-01  1.827135001E-09  1.98E-12  6.47E-06  9.34E+00  2.28E+09  6.4E-11
   350  1.000000957E-01  1.827111503E-09  1.98E-12  1.57E-09  9.47E+00  2.31E+09  6.4E-11