# Dense vectors using a NumPy backend.¶

This serves as a base class for dense vectors over Real Double Field and Complex Double Field

EXAMPLES:

sage: v = vector(CDF,[(1,-1), (2,pi), (3,5)])
sage: v
(1.0 - 1.0*I, 2.0 + 3.141592653589793*I, 3.0 + 5.0*I)
sage: type(v)
<type 'sage.modules.vector_complex_double_dense.Vector_complex_double_dense'>
sage: parent(v)
Vector space of dimension 3 over Complex Double Field
sage: v[0] = 5
sage: v
(5.0, 2.0 + 3.141592653589793*I, 3.0 + 5.0*I)
True
sage: v = vector(RDF, [1,2,3,4]); v
(1.0, 2.0, 3.0, 4.0)
True


AUTHORS:

• Jason Grout, Oct 2008: switch to numpy backend, factored out Vector_double_dense class
• Josh Kantor
• William Stein
class sage.modules.vector_double_dense.Vector_double_dense

Base class for vectors over the Real Double Field and the Complex Double Field. These are supposed to be fast vector operations using C doubles. Most operations are implemented using numpy which will call the underlying BLAS, if needed, on the system.

This class cannot be instantiated on its own. The numpy vector creation depends on several variables that are set in the subclasses.

EXAMPLES:

sage: v = vector(RDF, [1,2,3,4]); v
(1.0, 2.0, 3.0, 4.0)
sage: v*v
30.0

complex_vector()

Return the associated complex vector, i.e., this vector but with coefficients viewed as complex numbers.

EXAMPLES:

sage: v = vector(RDF,4,range(4)); v
(0.0, 1.0, 2.0, 3.0)
sage: v.complex_vector()
(0.0, 1.0, 2.0, 3.0)
sage: v = vector(RDF,0)
sage: v.complex_vector()
()

fft(direction='forward', algorithm='radix2', inplace=False)

This performs a fast Fourier transform on the vector.

INPUT:

• direction – ‘forward’ (default) or ‘backward’

The algorithm and inplace arguments are ignored.

This function is fastest if the vector’s length is a power of 2.

EXAMPLES:

sage: v = vector(CDF,[1+2*I,2,3*I,4])
sage: v.fft()
(7.0 + 5.0*I, 1.0 + 1.0*I, -5.0 + 5.0*I, 1.0 - 3.0*I)
sage: v.fft(direction='backward')
(1.75 + 1.25*I, 0.25 - 0.75*I, -1.25 + 1.25*I, 0.25 + 0.25*I)
sage: v.fft().fft(direction='backward')
(1.0 + 2.0*I, 2.0, 3.0*I, 4.0)
sage: v.fft().parent()
Vector space of dimension 4 over Complex Double Field
sage: v.fft(inplace=True)
sage: v
(7.0 + 5.0*I, 1.0 + 1.0*I, -5.0 + 5.0*I, 1.0 - 3.0*I)

sage: v = vector(RDF,4,range(4)); v
(0.0, 1.0, 2.0, 3.0)
sage: v.fft()
(6.0, -2.0 + 2.0*I, -2.0, -2.0 - 2.0*I)
sage: v.fft(direction='backward')
(1.5, -0.5 - 0.5*I, -0.5, -0.5 + 0.5*I)
sage: v.fft().fft(direction='backward')
(0.0, 1.0, 2.0, 3.0)
sage: v.fft().parent()
Vector space of dimension 4 over Complex Double Field
sage: v.fft(inplace=True)
Traceback (most recent call last):
...
ValueError: inplace can only be True for CDF vectors

inv_fft(algorithm='radix2', inplace=False)

This performs the inverse fast Fourier transform on the vector.

The Fourier transform can be done in place using the keyword inplace=True

This will be fastest if the vector’s length is a power of 2.

EXAMPLES:

sage: v = vector(CDF,[1,2,3,4])
sage: w = v.fft()
sage: max(v - w.inv_fft()) < 1e-12
True

mean()

Calculate the arithmetic mean of the vector.

EXAMPLES:

sage: v = vector(RDF, range(9))
sage: w = vector(CDF, [k+(9-k)*I for k in range(9)])
sage: v.mean()
4.0
sage: w.mean()
4.0 + 5.0*I

norm(p=2)

Returns the norm (or related computations) of the vector.

INPUT:

• p - default: 2 - controls which norm is computed, allowable values are any real number and positive and negative infinity. See output discussion for specifics.

OUTPUT:

Returned value is a double precision floating point value in RDF (or an integer when p=0). The default value of p = 2 is the “usual” Euclidean norm. For other values:

• p = Infinity or p = oo: the maximum of the absolute values of the entries, where the absolute value of the complex number $$a+bi$$ is $$\sqrt{a^2+b^2}$$.

• p = -Infinity or p = -oo: the minimum of the absolute values of the entries.

• p = 0 : the number of nonzero entries in the vector.

• p is any other real number: for a vector $$\vec{x}$$ this method computes

$\left(\sum_i x_i^p\right)^{1/p}$

For p < 0 this function is not a norm, but the above computation may be useful for other purposes.

ALGORITHM:

Computation is performed by the norm() function of the SciPy/NumPy library.

EXAMPLES:

First over the reals.

sage: v = vector(RDF, range(9))
sage: v.norm()
14.28285685...
sage: v.norm(p=2)
14.28285685...
sage: v.norm(p=6)
8.744039097...
sage: v.norm(p=Infinity)
8.0
sage: v.norm(p=-oo)
0.0
sage: v.norm(p=0)
8.0
sage: v.norm(p=0.3)
4099.153615...


And over the complex numbers.

sage: w = vector(CDF, [3-4*I, 0, 5+12*I])
sage: w.norm()
13.9283882...
sage: w.norm(p=2)
13.9283882...
sage: w.norm(p=0)
2.0
sage: w.norm(p=4.2)
13.0555695...
sage: w.norm(p=oo)
13.0


Negative values of p are allowed and will provide the same computation as for positive values. A zero entry in the vector will raise a warning and return zero.

sage: v = vector(CDF, range(1,10))
sage: v.norm(p=-3.2)
0.953760808...
sage: w = vector(CDF, [-1,0,1])
sage: w.norm(p=-1.6)
doctest:...: RuntimeWarning: divide by zero encountered in power
0.0


Return values are in RDF, or an integer when p = 0.

sage: v = vector(RDF, [1,2,4,8])
sage: v.norm() in RDF
True
sage: v.norm(p=0) in ZZ
True


Improper values of p are caught.

sage: w = vector(CDF, [-1,0,1])
sage: w.norm(p='junk')
Traceback (most recent call last):
...
ValueError: vector norm 'p' must be +/- infinity or a real number, not junk

numpy(dtype=None)

Return numpy array corresponding to this vector.

INPUT:

• dtype – if specified, the numpy dtype
of the returned array.

EXAMPLES:

sage: v = vector(CDF,4,range(4))
sage: v.numpy()
array([0.+0.j, 1.+0.j, 2.+0.j, 3.+0.j])
sage: v = vector(CDF,0)
sage: v.numpy()
array([], dtype=complex128)
sage: v = vector(RDF,4,range(4))
sage: v.numpy()
array([0., 1., 2., 3.])
sage: v = vector(RDF,0)
sage: v.numpy()
array([], dtype=float64)


A numpy dtype may be requested manually:

sage: import numpy
sage: v = vector(CDF, 3, range(3))
sage: v.numpy()
array([0.+0.j, 1.+0.j, 2.+0.j])
sage: v.numpy(dtype=numpy.float64)
array([0., 1., 2.])
sage: v.numpy(dtype=numpy.float32)
array([0., 1., 2.], dtype=float32)

prod()

Return the product of the entries of self.

EXAMPLES:

sage: v = vector(RDF, range(9))
sage: w = vector(CDF, [k+(9-k)*I for k in range(9)])
sage: v.prod()
0.0
sage: w.prod()
57204225.0*I

standard_deviation(population=True)

Calculate the standard deviation of entries of the vector.

INPUT:
population – If False, calculate the sample standard deviation.

EXAMPLES:

sage: v = vector(RDF, range(9))
sage: w = vector(CDF, [k+(9-k)*I for k in range(9)])
sage: v.standard_deviation()
2.7386127875258306
sage: v.standard_deviation(population=False)
2.581988897471611
sage: w.standard_deviation()
3.872983346207417
sage: w.standard_deviation(population=False)
3.6514837167011076

stats_kurtosis()

Compute the kurtosis of a dataset.

Kurtosis is the fourth central moment divided by the square of the variance. Since we use Fisher’s definition, 3.0 is subtracted from the result to give 0.0 for a normal distribution. (Paragraph from the scipy.stats docstring.)

EXAMPLES:

sage: v = vector(RDF, range(9))
sage: w = vector(CDF, [k+(9-k)*I for k in range(9)])
sage: v.stats_kurtosis()  # rel tol 5e-15
-1.2300000000000000
sage: w.stats_kurtosis()  # rel tol 5e-15
-1.2300000000000000

sum()

Return the sum of the entries of self.

EXAMPLES:

sage: v = vector(RDF, range(9))
sage: w = vector(CDF, [k+(9-k)*I for k in range(9)])
sage: v.sum()
36.0
sage: w.sum()
36.0 + 45.0*I

variance(population=True)

Calculate the variance of entries of the vector.

INPUT:

• population – If False, calculate the sample variance.

EXAMPLES:

sage: v = vector(RDF, range(9))
sage: w = vector(CDF, [k+(9-k)*I for k in range(9)])
sage: v.variance()
7.5
sage: v.variance(population=False)
6.666666666666667
sage: w.variance()
15.0
sage: w.variance(population=False)
13.333333333333334

zero_at(eps)

Returns a copy with small entries replaced by zeros.

This is useful for modifying output from algorithms which have large relative errors when producing zero elements, e.g. to create reliable doctests.

INPUT:

• eps - cutoff value

OUTPUT:

A modified copy of the vector. Elements smaller than or equal to eps are replaced with zeroes. For complex vectors, the real and imaginary parts are considered individually.

EXAMPLES:

sage: v = vector(RDF, [1.0, 2.0, 10^-10, 3.0])
sage: v.zero_at(1e-8)
(1.0, 2.0, 0.0, 3.0)
sage: v.zero_at(1e-12)
(1.0, 2.0, 1e-10, 3.0)


For complex numbers the real and imaginary parts are considered separately.

sage: w = vector(CDF, [10^-6 + 5*I, 5 + 10^-6*I, 5 + 5*I, 10^-6 + 10^-6*I])
sage: w.zero_at(1.0e-4)
(5.0*I, 5.0, 5.0 + 5.0*I, 0.0)
sage: w.zero_at(1.0e-8)
(1e-06 + 5.0*I, 5.0 + 1e-06*I, 5.0 + 5.0*I, 1e-06 + 1e-06*I)