Dense matrices over the Real Double Field using NumPy

EXAMPLES:

sage: b = Mat(RDF,2,3).basis()
sage: b[0,0]
[1.0 0.0 0.0]
[0.0 0.0 0.0]

>>> from sage.all import *
>>> b = Mat(RDF,Integer(2),Integer(3)).basis()
>>> b[Integer(0),Integer(0)]
[1.0 0.0 0.0]
[0.0 0.0 0.0]

b = Mat(RDF,2,3).basis()
b[0,0]

We deal with the case of zero rows or zero columns:

sage: m = MatrixSpace(RDF,0,3)
sage: m.zero_matrix()
[]

>>> from sage.all import *
>>> m = MatrixSpace(RDF,Integer(0),Integer(3))
>>> m.zero_matrix()
[]

m = MatrixSpace(RDF,0,3)
m.zero_matrix()

AUTHORS:

  • Jason Grout (2008-09): switch to NumPy backend, factored out the Matrix_double_dense class

  • Josh Kantor

  • William Stein: many bug fixes and touch ups.

class sage.matrix.matrix_real_double_dense.Matrix_real_double_dense[source]

Bases: Matrix_double_dense

Class that implements matrices over the real double field. These are supposed to be fast matrix operations using C doubles. Most operations are implemented using numpy which will call the underlying BLAS on the system.

EXAMPLES:

sage: m = Matrix(RDF, [[1,2],[3,4]])
sage: m**2
[ 7.0 10.0]
[15.0 22.0]
sage: n = m^(-1); n     # rel tol 1e-15                                         # needs scipy
[-1.9999999999999996  0.9999999999999998]
[ 1.4999999999999998 -0.4999999999999999]

>>> from sage.all import *
>>> m = Matrix(RDF, [[Integer(1),Integer(2)],[Integer(3),Integer(4)]])
>>> m**Integer(2)
[ 7.0 10.0]
[15.0 22.0]
>>> n = m**(-Integer(1)); n     # rel tol 1e-15                                         # needs scipy
[-1.9999999999999996  0.9999999999999998]
[ 1.4999999999999998 -0.4999999999999999]

m = Matrix(RDF, [[1,2],[3,4]])
m**2
n = m^(-1); n     # rel tol 1e-15                                         # needs scipy

To compute eigenvalues, use the method left_eigenvectors() or right_eigenvectors().

sage: p,e = m.right_eigenvectors()                                              # needs scipy

>>> from sage.all import *
>>> p,e = m.right_eigenvectors()                                              # needs scipy

p,e = m.right_eigenvectors()                                              # needs scipy

The result is a pair (p,e), where p is a diagonal matrix of eigenvalues and e is a matrix whose columns are the eigenvectors.

To solve a linear system \(Ax = b\) where A = [[1,2],[3,4]] and \(b = [5,6]\):

sage: b = vector(RDF,[5,6])
sage: m.solve_right(b)  # rel tol 1e-15                                         # needs scipy
(-3.9999999999999987, 4.499999999999999)

>>> from sage.all import *
>>> b = vector(RDF,[Integer(5),Integer(6)])
>>> m.solve_right(b)  # rel tol 1e-15                                         # needs scipy
(-3.9999999999999987, 4.499999999999999)

b = vector(RDF,[5,6])
m.solve_right(b)  # rel tol 1e-15                                         # needs scipy

See the methods QR(), LU(), and SVD() for QR, LU, and singular value decomposition.