1> help lugauss
`lugauss' is a function from the file C:\Documents and Settings\rudiger.achilles\desktop\lugauss.m

LUGAUSS   Fattorizzazione LU senza pivoting
   A = LUGAUSS(A) calcola la fattorizzazione
   LU di Gauss della matrice A, memorizzando nella
   parte triangolare inferiore stretta di A la
   matrice L (gli elementi diagonali di L sono tutti
   uguali a 1) ed in quella superiore il fattore U

Additional help for built-in functions and operators is
available in the on-line version of the manual.  Use the command
`doc <topic>' to search the manual index.

Help and information about Octave is also available on the WWW
at http://www.octave.org and via the help@octave.org
mailing list.


2> % Quarteroni-Saleri, p. 146, Esempio 5.8


2> A = [1 1+0.5e-15 3; 2 2 20; 3 6 4]
A =

    1.0000    1.0000    3.0000
    2.0000    2.0000   20.0000
    3.0000    6.0000    4.0000



3> B = lugauss(A)
B =

  1.0000e+000  1.0000e+000  3.0000e+000
  2.0000e+000  -8.8818e-016  1.4000e+001
  3.0000e+000  -3.3777e+015  4.7288e+016



4> L = tril(B,-1) + eye(3)
L =

  1.0000e+000  0.0000e+000  0.0000e+000
  2.0000e+000  1.0000e+000  0.0000e+000
  3.0000e+000  -3.3777e+015  1.0000e+000



5> U = triu(B)
U =

  1.0000e+000  1.0000e+000  3.0000e+000
  0.0000e+000  -8.8818e-016  1.4000e+001
  0.0000e+000  0.0000e+000  4.7288e+016



6> A - L * U 
ans =

   0   0   0
   0   0   0
   0   0   6



7> help lu
`lu' is a function from the file C:\Programmi\3.2.4_gcc-4.4.0\libexec\octave\3.2.4\oct\i686-pc-mingw32\lu.oct

 -- Loadable Function: [L, U, P] = lu (A)
 -- Loadable Function: [L, U, P, Q] = lu (S)
 -- Loadable Function: [L, U, P, Q, R] = lu (S)
 -- Loadable Function: [...] = lu (S, THRES)
 -- Loadable Function: Y = lu (...)
 -- Loadable Function: [...] = lu (..., 'vector')
     Compute the LU decomposition of A.  If A is full subroutines from
     LAPACK are used and if A is sparse then UMFPACK is used.  The
     result is returned in a permuted form, according to the optional
     return value P.  For example, given the matrix `a = [1, 2; 3, 4]',

          [l, u, p] = lu (a)

     returns

          l =

           1.00000  0.00000
           0.33333  1.00000

          u =

           3.00000  4.00000
           0.00000  0.66667

          p =

           0  1
           1  0

     The matrix is not required to be square.

     Called with two or three output arguments and a spare input matrix,
     then "lu" does not attempt to perform sparsity preserving column
     permutations.  Called with a fourth output argument, the sparsity
     preserving column transformation Q is returned, such that `P * A *
     Q = L * U'.

     Called with a fifth output argument and a sparse input matrix, then
     "lu" attempts to use a scaling factor R on the input matrix such
     that `P * (R \ A) * Q = L * U'.  This typically leads to a sparser
     and more stable factorization.

     An additional input argument THRES, that defines the pivoting
     threshold can be given.  THRES can be a scalar, in which case it
     defines UMFPACK pivoting tolerance for both symmetric and
     unsymmetric cases.  If THRES is a two element vector, then the
     first element defines the pivoting tolerance for the unsymmetric
     UMFPACK pivoting strategy and the second the symmetric strategy.
     By default, the values defined by `spparms' are used and are by
     default `[0.1, 0.001]'.

     Given the string argument 'vector', "lu" returns the values of P Q
     as vector values, such that for full matrix, `A (P,:) = L * U',
     and `R(P,:) * A (:, Q) = L * U'.

     With two output arguments, returns the permuted forms of the upper
     and lower triangular matrices, such that `A = L * U'.  With one
     output argument Y, then the matrix returned by the LAPACK routines
     is returned.  If the input matrix is sparse then the matrix L is
     embedded into U to give a return value similar to the full case.
     For both full and sparse matrices, "lu" looses the permutation
     information.


Additional help for built-in functions and operators is
available in the on-line version of the manual.  Use the command
`doc <topic>' to search the manual index.

Help and information about Octave is also available on the WWW
at http://www.octave.org and via the help@octave.org
mailing list.



8> % qui c'e' un errore: ho voluto calcolare lu(A)

8> [l, u, p] = lu(B)
l =

   1.00000   0.00000   0.00000
   0.66667   1.00000   0.00000
   0.33333   0.50000   1.00000

u =

  3.0000e+000  -3.3777e+015  4.7288e+016
  0.0000e+000  2.2518e+015  -3.1525e+016
  0.0000e+000  0.0000e+000  1.0000e+001

p =

Permutation Matrix

   0   0   1
   0   1   0
   1   0   0



9> p * A - l * u
ans =

  0.0000e+000  3.3777e+015  -4.7288e+016
  0.0000e+000  1.8750e+000  7.0840e+000
  0.0000e+000  -6.2500e-002  5.4199e-001



10> A
A =

    1.0000    1.0000    3.0000
    2.0000    2.0000   20.0000
    3.0000    6.0000    4.0000



11> p * A
ans =

    3.0000    6.0000    4.0000
    2.0000    2.0000   20.0000
    1.0000    1.0000    3.0000



12> l
l =

   1.00000   0.00000   0.00000
   0.66667   1.00000   0.00000
   0.33333   0.50000   1.00000



13> u
u =

  3.0000e+000  -3.3777e+015  4.7288e+016
  0.0000e+000  2.2518e+015  -3.1525e+016
  0.0000e+000  0.0000e+000  1.0000e+001



14> l*u
ans =

  3.0000e+000  -3.3777e+015  4.7288e+016
  2.0000e+000  1.2500e-001  1.2916e+001
  1.0000e+000  1.0625e+000  2.4580e+000



15> U
U =

  1.0000e+000  1.0000e+000  3.0000e+000
  0.0000e+000  -8.8818e-016  1.4000e+001
  0.0000e+000  0.0000e+000  4.7288e+016



16> p * A
ans =

    3.0000    6.0000    4.0000
    2.0000    2.0000   20.0000
    1.0000    1.0000    3.0000



17> % Quarteroni-Saleri, p. 147, Esempio 5.9


17> % sistema lineare mal condizionato


17> a = hilb(13);


18> cond(a)
ans = 1.9487e+018


19> x = ones(13,1);


20> b = a * x
b =

   3.18013
   2.25156
   1.81823
   1.54740
   1.35622
   1.21177
   1.09774
   1.00488
   0.92750
   0.86184
   0.80532
   0.75608
   0.71275



21> % da completare la prossima volta, cioe' il 25/11/2011


21> quit