[NumAn] Les 4 -- Sage

M = random_matrix(RDF, 4, 4)

M

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 0.0550985530275 & 0.0637216162706 & -0.395332726592 & 0.289319416962 \\ -0.903052940785 & 0.340530518985 & -0.633020553187 & 0.0183468791879 \\ -0.318671408701 & -0.0579825863608 & 0.465085789116 & -0.0659918352696 \\ -0.200402478869 & -0.306615926381 & 0.375686874882 & 0.767268558979 \end{array}\right)$

P, L, U = M.LU() P, L, U

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left(\begin{array}{rrrr} 0.0 & 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 1.0 \\ 0.0 & 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 & 0.0 \end{array}\right), \left(\begin{array}{rrrr} 1.0 & 0.0 & 0.0 & 0.0 \\ 0.221916644992 & 1.0 & 0.0 & 0.0 \\ 0.352882311002 & 0.466134555923 & 1.0 & 0.0 \\ -0.0610136466413 & -0.221093331796 & -0.714132157234 & 1.0 \end{array}\right), \left(\begin{array}{rrrr} -0.903052940785 & 0.340530518985 & -0.633020553187 & 0.0183468791879 \\ 0.0 & -0.382185316672 & 0.516164672256 & 0.763197081103 \\ 0.0 & 0.0 & 0.447865354551 & -0.428218656879 \\ 0.0 & 0.0 & 0.0 & 0.15337189924 \end{array}\right)\right)$

P^-1*L*U - M

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 6.93889390391 \times 10^{-18} & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & -1.38777878078 \times 10^{-17} & 0.0 & 0.0 \\ 0.0 & 0.0 & 5.55111512313 \times 10^{-17} & 0.0 \end{array}\right)$

Q, R = M.QR() Q, R

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left(\begin{array}{rrrr} -0.0562273140901 & -0.188464848629 & 0.562315829185 & 0.803193873376 \\ 0.921553081734 & -0.319434625008 & 0.175741932455 & -0.133477380365 \\ 0.325199781193 & 0.325711384843 & -0.677610241089 & 0.573586573471 \\ 0.20450796808 & 0.86968650384 & 0.440181829015 & -0.0897877131653 \end{array}\right), \left(\begin{array}{rrrr} -0.979925040331 & 0.22867272938 & -0.333056787929 & 0.136091972904 \\ 0.0 & -0.406331844937 & 0.754928546703 & 0.585401649913 \\ 0.0 & 0.0 & -0.483326463226 & 0.548367624896 \\ 0.0 & 0.0 & 0.0 & 0.123187369817 \end{array}\right)\right)$

Q*R - M

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 0.0 & 6.93889390391 \times 10^{-17} & 1.11022302463 \times 10^{-16} & 0.0 \\ 0.0 & 0.0 & 0.0 & 6.93889390391 \times 10^{-18} \\ 5.55111512313 \times 10^{-17} & 0.0 & -1.11022302463 \times 10^{-16} & 4.16333634234 \times 10^{-17} \\ 2.77555756156 \times 10^{-17} & -5.55111512313 \times 10^{-17} & 0.0 & 0.0 \end{array}\right)$

Q * Q.transpose()

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 1.0 & -9.71445146547 \times 10^{-17} & 0.0 & -1.80411241502 \times 10^{-16} \\ -9.71445146547 \times 10^{-17} & 1.0 & 2.77555756156 \times 10^{-17} & -4.68375338514 \times 10^{-17} \\ 0.0 & 2.77555756156 \times 10^{-17} & 1.0 & 6.93889390391 \times 10^{-18} \\ -1.80411241502 \times 10^{-16} & -4.68375338514 \times 10^{-17} & 6.93889390391 \times 10^{-18} & 1.0 \end{array}\right)$

M * x = b Q*R *x = b R*x = Q.transpose() * b

Traceback (click to the left of this block for traceback)
...
SyntaxError: can't assign to operator

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_21.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("TSAqIHggID0gYgpRKlIgKnggID0gYgpSKnggPSBRLnRyYW5zcG9zZSgpICogYg=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/var/sage/tmpAUpvtc/___code___.py", line 2
    M * x  = b
SyntaxError: can't assign to operator

P = M*M.transpose() L = (P).cholesky() L

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 0.497081466836 & 0.0 & 0.0 & 0.0 \\ 0.457679695509 & 1.0597389175 & 0.0 & 0.0 \\ -0.451051678654 & 0.168767888772 & 0.306011345391 & 0.0 \\ 0.0862724309685 & -0.176141216781 & 0.896615113188 & 0.147067425396 \end{array}\right)$

L*L.transpose() - P

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -2.77555756156 \times 10^{-17} & 2.77555756156 \times 10^{-17} & -2.77555756156 \times 10^{-17} & 6.93889390391 \times 10^{-18} \\ 2.77555756156 \times 10^{-17} & -2.22044604925 \times 10^{-16} & 6.93889390391 \times 10^{-18} & 0.0 \\ -2.77555756156 \times 10^{-17} & 6.93889390391 \times 10^{-18} & 0.0 & 0.0 \\ 6.93889390391 \times 10^{-18} & 0.0 & 0.0 & 0.0 \end{array}\right)$

M = random_matrix(RDF, 4, 4) b = random_vector(RDF, 4)

LUmatrix = list(map(list, M)) for k in range(len(LUmatrix) - 1): for i in range(k+1, len(LUmatrix)): LUmatrix[i][k] /= LUmatrix[k][k] for j in range(k+1, len(LUmatrix)): for i in range(k+1, len(LUmatrix)): LUmatrix[i][j] -= LUmatrix[i][k] * LUmatrix[k][j]

x = list(b) for k in range(1, len(x)): for j in range(k): x[k] -= LUmatrix[k][j] * x[j] for k in range(len(x) - 1, -1, -1): for j in range(k+1, len(x)): x[k] -= LUmatrix[k][j] * x[j] x[k] /= LUmatrix[k][k]

M.solve_right(b)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(-0.599136465947,\,0.11159537258,\,-0.272333571707,\,-1.00683950818\right)$

x

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[-0.599136465947, 0.11159537258, -0.272333571707, -1.00683950818\right]$

[NumAn] Les 4

1887 days ago by Toon.Baeyens