Sharedsupport / 2015-01-21-length.sagewsOpen in CoCalc
Authors: Harald Schilly, ℏal Snyder, William A. Stein
License: GNU General Public License v3.0
Description: Examples for support purposes.
v = vector(RDF,[1,2,3]) %timeit v.norm()
625 loops, best of 3: 61.7 µs per loop
File: /usr/local/sage/sage-6.4/src/sage/modules/vector_double_dense.pyx Source: def norm(self, p=2): r""" 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 .. math:: \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 """ global numpy if numpy is None: import numpy import sage.rings.infinity import sage.rings.integer if p == sage.rings.infinity.Infinity: p = numpy.inf elif p == -sage.rings.infinity.Infinity: p = -numpy.inf else: try: p = RDF(p) except Exception: raise ValueError("vector norm 'p' must be +/- infinity or a real number, not %s" % p) n = numpy.linalg.norm(self._vector_numpy, ord=p) # p = 0 returns integer *count* of non-zero entries return RDF(n)
import numpy %timeit numpy.linalg.norm(v.numpy(), ord=2r)
625 loops, best of 3: 21 µs per loop
numpy.linalg.norm(v.numpy(), ord=2r)
w = v.numpy()
%timeit numpy.linalg.norm(w)
625 loops, best of 3: 11.1 µs per loop