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

 --------------------------------------------------------------------
 Least-Squares Test Problem      P( 2000 1000   40    2    1.00E-02 )
 Condition no. =  9.8749E+01     Residual function =  3.165162796E+01
 --------------------------------------------------------------------


 Enter Acheck.     Test of Aprod for LSQR and CRAIG
 Aprod seems OK.   Relative error =   7.6E-16


 Enter LSQR.       Least-squares solution of  Ax = b
 The matrix  A  has   2000 rows   and   1000 columns
 damp   =  1.00000000000000E-02   wantse =         F
 atol   =  3.18E-16               conlim =  9.87E+04
 btol   =  3.18E-16               itnlim =     12200


   Itn       x(1)           Function     Compatible   LS     Norm Abar Cond Abar alfa_opt
     0  0.000000000E+00  1.251026233E+03  1.00E+00  6.62E-04
     1 -1.569382661E+01  4.513015121E+02  3.61E-01  7.04E-01  8.88E-01  1.00E+00  5.5E-01
     2  2.897013496E+00  2.389170081E+02  1.91E-01  4.36E-01  1.18E+00  2.12E+00  2.9E-01
     3  6.912476123E+00  1.508301251E+02  1.21E-01  3.14E-01  1.39E+00  3.40E+00  1.9E-01
     4  1.925468458E+00  1.061006594E+02  8.48E-02  2.39E-01  1.56E+00  4.82E+00  1.4E-01
     5 -4.031144634E+00  8.068847228E+01  6.45E-02  1.86E-01  1.70E+00  6.39E+00  1.1E-01
     6 -8.164502129E+00  6.526230465E+01  5.22E-02  1.45E-01  1.83E+00  8.09E+00  9.2E-02
     7 -1.026456074E+01  5.551640158E+01  4.44E-02  1.13E-01  1.93E+00  9.92E+00  7.8E-02
     8 -1.083036972E+01  4.920410096E+01  3.93E-02  8.72E-02  2.02E+00  1.19E+01  6.8E-02
     9 -1.039855535E+01  4.504692088E+01  3.60E-02  6.64E-02  2.09E+00  1.40E+01  6.1E-02
    10 -9.380373468E+00  4.227482078E+01  3.38E-02  5.00E-02  2.15E+00  1.63E+01  5.6E-02
    20  8.956177896E-01  3.662630556E+01  2.93E-02  1.58E-03  2.55E+00  6.45E+01  2.8E-02
    30  4.676353374E-01  3.654706202E+01  2.92E-02  4.86E-05  3.15E+00  2.51E+02  1.6E-02
    40  1.003079085E-01  3.654666958E+01  2.92E-02  3.77E-06  3.66E+00  5.51E+02  1.2E-02
    50  9.999999951E-02  3.654666951E+01  2.92E-02  1.52E-11  4.14E+00  6.26E+02  1.1E-02
    59  1.000000000E-01  3.654666951E+01  2.92E-02  7.71E-16  4.50E+00  6.86E+02  1.1E-02
    60  1.000000000E-01  3.654666951E+01  2.92E-02  1.60E-15  4.50E+00  6.90E+02  1.1E-02
    61  1.000000000E-01  3.654666951E+01  2.92E-02  3.54E-16  4.55E+00  6.99E+02  1.1E-02
    62  1.000000000E-01  3.654666951E+01  2.92E-02  1.08E-16  4.57E+00  7.01E+02  1.1E-02

 Exit  LSQR.       istop  = 3               itn    =      62
 Exit  LSQR.       Anorm  = 4.56805E+00     Acond  = 7.01224E+02
 Exit  LSQR.       bnorm  = 1.25103E+03     xnorm  = 1.82711E+03
 Exit  LSQR.       rnorm  = 3.65467E+01     Arnorm = 1.80399E-14
 Exit  LSQR.       max dx = 1.3E+03 occurred at itn        1
 Exit  LSQR.              = 7.2E-01*xnorm
 Exit  LSQR.       A damped least-squares solution was found, given atol


 Enter xcheck.     Does x solve Ax = b, etc?
    damp            = 1.000E-02
    norm(x)         = 1.827E+03
    norm(r)         = 3.16516280E+01 = rho1
    norm(A'r)       = 1.827E-01      = sigma1

    norm(s)         = 3.165E+03
    norm(x,s)       = 3.655E+03
    norm(rbar)      = 3.65466695E+01 = rho2
    norm(Abar'rbar) = 5.685E-13      = sigma2

    inform          = 3
    tol             = 1.490E-08
    test1           = 3.298E-03 (Ax = b)
    test2           = 1.264E-03 (least-squares)
    test3           = 3.405E-15 (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  =  5.77E-15


 --------------------------------------------------------------------
 Least-Squares Test Problem      P( 2000 1000   40    3    1.00E-03 )
 Condition no. =  9.9796E+02     Residual function =  3.162364242E+01
 --------------------------------------------------------------------


 Enter Acheck.     Test of Aprod for LSQR and CRAIG
 Aprod seems OK.   Relative error =   4.6E-16


 Enter LSQR.       Least-squares solution of  Ax = b
 The matrix  A  has   2000 rows   and   1000 columns
 damp   =  1.00000000000000E-03   wantse =         F
 atol   =  3.18E-16               conlim =  9.98E+05
 btol   =  3.18E-16               itnlim =     12200


   Itn       x(1)           Function     Compatible   LS     Norm Abar Cond Abar alfa_opt
     0  0.000000000E+00  1.124761730E+03  1.00E+00  7.28E-04
     1 -2.024778976E+01  4.445508835E+02  3.95E-01  6.73E-01  8.91E-01  1.00E+00  5.8E-01
     2 -4.437036642E+00  2.497578091E+02  2.22E-01  4.11E-01  1.18E+00  2.15E+00  3.1E-01
     3  7.016901406E+00  1.633423201E+02  1.45E-01  2.93E-01  1.38E+00  3.47E+00  2.1E-01
     4  1.026269712E+01  1.165898904E+02  1.04E-01  2.22E-01  1.54E+00  4.95E+00  1.5E-01
     5  8.511232527E+00  8.829972244E+01  7.85E-02  1.73E-01  1.66E+00  6.61E+00  1.2E-01
     6  4.563956177E+00  6.998116189E+01  6.22E-02  1.35E-01  1.76E+00  8.45E+00  9.3E-02
     7  6.876673714E-02  5.763216488E+01  5.12E-02  1.05E-01  1.83E+00  1.05E+01  7.7E-02
     8 -4.113020748E+00  4.913193846E+01  4.37E-02  8.04E-02  1.89E+00  1.28E+01  6.5E-02
     9 -7.562594838E+00  4.324449567E+01  3.84E-02  5.99E-02  1.93E+00  1.54E+01  5.6E-02
    10 -1.009778237E+01  3.918723449E+01  3.48E-02  4.31E-02  1.96E+00  1.83E+01  4.9E-02
    20 -5.028362208E+00  3.179504338E+01  2.83E-02  1.16E-03  2.50E+00  9.38E+01  2.2E-02
    30  6.225162344E-01  3.168093575E+01  2.82E-02  1.29E-03  2.98E+00  3.23E+02  1.3E-02
    40  1.363094331E+00  3.167653023E+01  2.82E-02  1.93E-06  3.50E+00  1.01E+03  7.7E-03
    50  7.373757826E-01  3.167639280E+01  2.82E-02  1.69E-06  3.84E+00  2.26E+03  5.4E-03
    60  1.405348541E-01  3.167638076E+01  2.82E-02  8.50E-08  4.23E+00  4.54E+03  4.0E-03
    70  9.999935981E-02  3.167638071E+01  2.82E-02  1.13E-09  4.55E+00  6.62E+03  3.4E-03
    80  9.999999378E-02  3.167638071E+01  2.82E-02  3.45E-12  4.89E+00  7.13E+03  3.4E-03
    90  9.999999986E-02  3.167638071E+01  2.82E-02  2.45E-13  5.17E+00  7.56E+03  3.4E-03
    93  9.999999998E-02  3.167638071E+01  2.82E-02  4.61E-17  5.27E+00  7.72E+03  3.4E-03

 Exit  LSQR.       istop  = 3               itn    =      93
 Exit  LSQR.       Anorm  = 5.26803E+00     Acond  = 7.71629E+03
 Exit  LSQR.       bnorm  = 1.12476E+03     xnorm  = 1.82711E+03
 Exit  LSQR.       rnorm  = 3.16764E+01     Arnorm = 7.68724E-15
 Exit  LSQR.       max dx = 1.2E+03 occurred at itn        1
 Exit  LSQR.              = 6.3E-01*xnorm
 Exit  LSQR.       A damped least-squares solution was found, given atol


 Enter xcheck.     Does x solve Ax = b, etc?
    damp            = 1.000E-03
    norm(x)         = 1.827E+03
    norm(r)         = 3.16236424E+01 = rho1
    norm(A'r)       = 1.827E-03      = sigma1

    norm(s)         = 3.162E+04
    norm(x,s)       = 3.168E+04
    norm(rbar)      = 3.16763807E+01 = rho2
    norm(Abar'rbar) = 8.718E-13      = sigma2

    inform          = 3
    tol             = 1.490E-08
    test1           = 2.942E-03 (Ax = b)
    test2           = 1.097E-05 (least-squares)
    test3           = 5.225E-15 (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  =  1.72E-13


 --------------------------------------------------------------------
 Least-Squares Test Problem      P( 2000 1000   40    4    1.00E-04 )
 Condition no. =  9.9967E+03     Residual function =  3.162282452E+01
 --------------------------------------------------------------------


 Enter Acheck.     Test of Aprod for LSQR and CRAIG
 Aprod seems OK.   Relative error =   1.3E-16


 Enter LSQR.       Least-squares solution of  Ax = b
 The matrix  A  has   2000 rows   and   1000 columns
 damp   =  1.00000000000000E-04   wantse =         F
 atol   =  3.18E-16               conlim =  1.00E+07
 btol   =  3.18E-16               itnlim =     12200


   Itn       x(1)           Function     Compatible   LS     Norm Abar Cond Abar alfa_opt
     0  0.000000000E+00  1.036803666E+03  1.00E+00  7.88E-04
     1 -2.290522276E+01  4.288179779E+02  4.14E-01  6.51E-01  8.98E-01  1.00E+00  6.1E-01
     2 -1.229519069E+01  2.473115919E+02  2.39E-01  3.92E-01  1.18E+00  2.16E+00  3.2E-01
     3  2.912839266E-01  1.637963649E+02  1.58E-01  2.74E-01  1.37E+00  3.52E+00  2.1E-01
     4  8.372079894E+00  1.171217907E+02  1.13E-01  2.03E-01  1.51E+00  5.07E+00  1.5E-01
     5  1.186967936E+01  8.809260300E+01  8.50E-02  1.54E-01  1.61E+00  6.84E+00  1.2E-01
     6  1.189446970E+01  6.891456184E+01  6.65E-02  1.16E-01  1.68E+00  8.87E+00  9.0E-02
     7  9.555838382E+00  5.587301182E+01  5.39E-02  8.60E-02  1.73E+00  1.12E+01  7.3E-02
     8  5.773286086E+00  4.696784938E+01  4.53E-02  6.16E-02  1.76E+00  1.40E+01  6.0E-02
     9  1.300142613E+00  4.098617139E+01  3.95E-02  4.20E-02  1.79E+00  1.75E+01  5.0E-02
    10 -3.232562301E+00  3.709631214E+01  3.58E-02  2.71E-02  1.80E+00  2.18E+01  4.2E-02
    20 -1.047338372E+01  3.168716491E+01  3.06E-02  3.77E-04  2.43E+00  1.47E+02  1.7E-02
    30 -2.131856168E+00  3.162499006E+01  3.05E-02  3.92E-05  2.92E+00  6.22E+02  9.0E-03
    40  8.750540925E-01  3.162347281E+01  3.05E-02  5.62E-05  3.36E+00  2.15E+03  5.2E-03
    50  1.482699147E+00  3.162335684E+01  3.05E-02  2.11E-05  3.76E+00  6.18E+03  3.2E-03
    60  1.462228123E+00  3.162335497E+01  3.05E-02  7.67E-07  4.07E+00  8.00E+03  3.0E-03
    70  8.833865648E-01  3.162335253E+01  3.05E-02  6.61E-09  4.39E+00  2.06E+04  1.9E-03
    80  1.462338138E-01  3.162335236E+01  3.05E-02  1.21E-06  4.70E+00  4.86E+04  1.3E-03
    90  1.319365245E-01  3.162335236E+01  3.05E-02  6.04E-09  4.98E+00  5.20E+04  1.3E-03
   100  1.002811364E-01  3.162335236E+01  3.05E-02  2.25E-08  5.27E+00  7.56E+04  1.1E-03
   110  9.999998219E-02  3.162335236E+01  3.05E-02  1.20E-10  5.54E+00  7.98E+04  1.1E-03
   120  9.999997810E-02  3.162335236E+01  3.05E-02  1.64E-13  5.78E+00  8.33E+04  1.1E-03
   130  9.999999965E-02  3.162335236E+01  3.05E-02  4.45E-15  6.04E+00  8.71E+04  1.1E-03
   132  9.999999972E-02  3.162335236E+01  3.05E-02  1.62E-15  6.12E+00  8.83E+04  1.1E-03
   133  9.999999972E-02  3.162335236E+01  3.05E-02  2.02E-16  6.13E+00  8.83E+04  1.1E-03

 Exit  LSQR.       istop  = 3               itn    =     133
 Exit  LSQR.       Anorm  = 6.12613E+00     Acond  = 8.83042E+04
 Exit  LSQR.       bnorm  = 1.03680E+03     xnorm  = 1.82711E+03
 Exit  LSQR.       rnorm  = 3.16234E+01     Arnorm = 3.90657E-14
 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-04
    norm(x)         = 1.827E+03
    norm(r)         = 3.16228245E+01 = rho1
    norm(A'r)       = 1.827E-05      = sigma1

    norm(s)         = 3.162E+05
    norm(x,s)       = 3.162E+05
    norm(rbar)      = 3.16233524E+01 = rho2
    norm(Abar'rbar) = 6.106E-13      = sigma2

    inform          = 3
    tol             = 1.490E-08
    test1           = 2.586E-03 (Ax = b)
    test2           = 9.431E-08 (least-squares)
    test3           = 3.152E-15 (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  =  5.26E-12