pyacs.lib.glinalg module¶
Linear algebra for Geodesy problems
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pyacs.lib.glinalg.ls(G, d, verbose=False)[source]¶ Solve the least-squares (LS) problem m so that |Gx-d|**2 is minimum.
- Parameters
G – m x n model matrix as 2D numpy array
d – m 1D numpy observation vector
verbose – verbose mode
- Returns
x,chi2: m (1D numpy array of dim m), chi2 (chi-square)
- Note
solved through numpy.linalg.lstsq
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pyacs.lib.glinalg.lsw(G, d, std)[source]¶ Solve the least-squares (LS) with data uncertainties provided as a vector
- Parameters
G – m x n model matrix as 2D numpy array
d – m 1D numpy observation vector
std – standard deviation vector for d
- Note
the system is modified to be solved by ordinary LS by the change G<- (G.T/std).T and d<- d/std
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pyacs.lib.glinalg.lscov(G, d, cov, method='chol')[source]¶ Solve the least-squares (LS) problem with data covariance
- Parameters
G – m x n model matrix as 2D numpy array
d – m 1D numpy observation vector
cov – covariance matrix for d
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pyacs.lib.glinalg.lsw_full(G, d, std, verbose=False)[source]¶ Solve the least-squares (LS) with data uncertainties provided as a vector and returns the posterior covariance
- Parameters
G – m x n model matrix as 2D numpy array
d – m 1D numpy observation vector
std – standard deviation vector for d
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pyacs.lib.glinalg.lscov_full(G, d, cov, verbose=False)[source]¶ Solve the least-squares (LS) with data covariance and returns the posterior covariance
- Parameters
G – m x n model matrix as 2D numpy array
d – m 1D numpy observation vector
cov – covariance matrix for d
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pyacs.lib.glinalg.cov_to_invcov(M)[source]¶ Inverse a covariance matrix
- Parameters
cov – 2D numpy array covariance matrix
- Returns
2D numpy array inverse covariance matrix
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pyacs.lib.glinalg.corr_to_cov(corr, sigma_m)[source]¶ Correlation to covariance matrix
- Parameters
corr – correlation matrix
sigma_m – vector of standard deviation = sqrt(diag(Cov))
:return covariance matrix
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pyacs.lib.glinalg.cov_to_corr(Cov)[source]¶ Covariance to correlation transformation
- Parameters
cov – covariance matrix
- Return corr,sigma_m
correlation matrix and standard deviation matrix
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pyacs.lib.glinalg.symmetrize(a, type_matrix)[source]¶ Extract the upper or lower triangle and make a symmetric matrix
- Parameters
a – numpy 2D array, must be a square matrix
type – ‘triu’ or ‘tril’, triangle used to form the matrix
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pyacs.lib.glinalg.dot_and_sum(LX, a)[source]¶ does a matrix by scalar product and sum it :param LX : list of arrays with the same dimension :param a : list of scalars
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pyacs.lib.glinalg.repeat_matrix_in_col(G, n)[source]¶ Repeat a matrix along a column
R = np.empty( ( n,G.shape[0], G.shape[1] ) ) R[:] = G return( R.reshape( n*G.shape[0], G.shape[1]) )
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pyacs.lib.glinalg.odot(a, G)[source]¶ Customize vector by matrix product.
Let’s assume we have a matrix made of equal dimension sub-matrices $G_1,cdots, G_n$ egin{equation} left[ egin{array}{c} G_1G_2
- dots
G_n end{array}
- ight]
end{equation} and a vector of scalars egin{equation} left[ egin{array}{c} a_1a_2
- dots
a_n end{array}
- ight]
end{equation} and we want egin{equation} left[ egin{array}{c} a_1 G_1a_2 G_2
- dots
a_n G_n end{array}
- ight]
end{equation} We do this with numpy broadcasting
- param a
1D numpy array of multipliers (scalars)
- param G
2-D numpy arrays of submatrix G_i.
- return
result of the multiplication
- note
if G.shape = (n,m), and a.shape=(l), then the shape of the G_i is (n/l,m)
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pyacs.lib.glinalg.dot(A, B)[source]¶ Matrix/matrix, matrix/vector and vector/vector multiplication :author : P. Rebischung :date:Created : 02-Aug-2013 :changes : :param A: Matrix or vector :param B : Matrix or vector :return: A*B
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pyacs.lib.glinalg.syminv(M)[source]¶ Invert a symmetric matrix :author : P. Rebischung :created : 02-Aug-2013 :param M : Matrix :return:Inverse matrix
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pyacs.lib.glinalg.sympinv(M, verbose=False)[source]¶ Pseudo-invert a positive semi-definite symmetric matrix :author: P. Rebischung :created : 02-Aug-2013 :param M : Matrix :return: Pseudo-inverse
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pyacs.lib.glinalg.make_normal_system(A, d, inv_Cd)[source]¶ Given the linear system A x = d with Cd the covariance matrix of d the associated normal system is A.T Cd-1 A x = A.T Cd-1 d. returns N= A.T Cd-1 A, Nd=A.T Cd-1 d
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pyacs.lib.glinalg.make_normal_system_tarantola(G, d, m0, inv_Cd, inv_Cm)[source]¶ returns Gt inv_Cd G + inv_Cm , Gt inv_Cd d + inv_Cm m0
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pyacs.lib.glinalg.matrix_from_pattern(pattern, structure)[source]¶ creates a matrix made of a pattern according to a structure
:param pattern : pattern 2D numpy array to be duplicated :param structure: 2D numpy array giving the structure as multiplicating factors :return: the block matrix as a 2D numpy array
>>> pattern = np.arange(6).reshape(2,3) >>> structure = np.arange(6).reshape(3,2)
>>> print(pattern) >>> print('--') >>> print(structure) >>> print('--') >>> print( matrix_from_pattern( pattern , structure ) )
[[0 1 2] [3 4 5]] – [[0 1]
[2 3] [4 5]]
– [[ 0 0 0 0 1 2]
[ 0 0 0 3 4 5] [ 0 2 4 0 3 6] [ 6 8 10 9 12 15] [ 0 4 8 0 5 10] [12 16 20 15 20 25]]