Differences between revisions 4 and 5
Revision 4 as of 2009-06-05 19:44:32
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Editor: Minh Nguyen
Comment: Summarize #6016, #3699
Revision 5 as of 2009-06-05 21:39:11
Size: 8404
Editor: Minh Nguyen
Comment: Summarize #6043
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 * FIXME: summarize #6081

 * FIXME: summarize #6043
 * Optimize the construction of Lagrange interpolation polynomials (Minh Van Nguyen) -- Rewrite the method {{{lagrange_polynomial()}}} in the class {{{PolynomialRing_field}}} of {{{sage/rings/polynomial/polynomial_ring.py}}} for generating the {{{n}}}-th Lagrange interpolation polynomial. The method now provides two new options:

  * {{{algorithm}}} --- (default: {{{divided_difference}}}) If {{{algorithm="divided_difference"}}}, then use the method of divided difference. If {{{algorithm="neville"}}}, then use a variant of Neville's method to recursively generate the {{{n}}}-th Lagrange interpolation polynomial. This adaptation of Neville's method is more memory efficient than the original Neville's method, since the former doesn't generate the full Neville table resulting from Neville's recursive procedure. Instead the adaptation only keeps track of the current and previous rows of the said table.

  * {{{previous_row}}} --- (default: {{{None}}}) This is only relevant if used together with {{{algorithm="neville"}}}. Here "previous row" refers to the last row in the Neville table that was obtained from a previous computation of an {{{n}}}-th Lagrange interpolation polynomial using Neville's method. If the last row is provided, then use a memory efficient variant of Neville's method to recursively generate a better interpolation polynomial from the results of previous computation.

 There's also the new method {{{divided_difference()}}} to compute the Newton divided-difference coefficients of the {{{n}}}-th Lagrange interpolation polynomial. The following are some timing statistics obtained using sage.math. When the results of previous computations are fed to {{{lagrange_polynomial}}} in order to produce better interpolation polynomials, we can gain an efficiency of up to 42%.
 {{{
# BEFORE

# using the definition of Lagrange interpolation polynomial
sage: R = PolynomialRing(QQ, 'x')
sage: %timeit R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)])
1000 loops, best of 3: 1.71 ms per loop
sage: R = PolynomialRing(GF(2**3,'a'), 'x')
sage: a = R.base_ring().gen()
sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)])")
625 loops, best of 3: 233 µs per loop

# without using precomputed values to generate successively better interpolation polynomials

sage: R = PolynomialRing(QQ, 'x')
sage: timeit("R.lagrange_polynomial([(0,1),(2,2)])");
625 loops, best of 3: 571 µs per loop
sage: # add two more points
sage: timeit("R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)])");
125 loops, best of 3: 2.29 ms per loop
sage:
sage: R = PolynomialRing(GF(2**3,'a'), 'x')
sage: a = R.base_ring().gen()
sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1)])")
625 loops, best of 3: 76.1 µs per loop
sage: # add another point
sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)])")
625 loops, best of 3: 229 µs per loop
sage:
sage: R = PolynomialRing(QQ, 'x')
sage: points = [(random(), random()) for i in xrange(100)]
sage: time R.lagrange_polynomial(points);
CPU times: user 1.21 s, sys: 0.00 s, total: 1.21 s
Wall time: 1.21 s
sage: # add three more points
sage: for i in xrange(3): points.append((random(), random()))
....:
sage: time R.lagrange_polynomial(points);
CPU times: user 1.28 s, sys: 0.01 s, total: 1.29 s
Wall time: 1.29 s
sage: # add another 100 points
sage: for i in xrange(100): points.append((random(), random()))
....:
sage: time R.lagrange_polynomial(points);
CPU times: user 5.87 s, sys: 0.02 s, total: 5.89 s
Wall time: 5.89 s


# AFTER

# using the method of divided-difference
sage: R = PolynomialRing(QQ, 'x')
sage: %timeit R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)])
1000 loops, best of 3: 827 µs per loop
sage: R = PolynomialRing(GF(2**3,'a'), 'x')
sage: a = R.base_ring().gen()
sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)])")
625 loops, best of 3: 111 µs per loop

# using precomputed values to generate successively better interpolation polynomials

sage: R = PolynomialRing(QQ, 'x')
sage: timeit("R.lagrange_polynomial([(0,1),(2,2)], neville=True)");
625 loops, best of 3: 332 µs per loop
sage: p = R.lagrange_polynomial([(0,1),(2,2)], neville=True);
sage: # add two more points
sage: timeit("R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)], neville=True, previous_row=p)");
625 loops, best of 3: 1.41 ms per loop
sage:
sage: R = PolynomialRing(GF(2**3,'a'), 'x')
sage: a = R.base_ring().gen()
sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1)], neville=True)");
625 loops, best of 3: 36.4 µs per loop
sage: p = R.lagrange_polynomial([(a^2+a,a),(a,1)], neville=True);
sage: # add another point
sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)], neville=True, previous_row=p)");
625 loops, best of 3: 131 µs per loop
sage:
sage: R = PolynomialRing(QQ, 'x')
sage: points = [(random(), random()) for i in xrange(100)]
sage: time R.lagrange_polynomial(points, neville=True);
CPU times: user 1.26 s, sys: 0.00 s, total: 1.26 s
Wall time: 1.26 s
sage: p = R.lagrange_polynomial(points, neville=True);
sage: # add three more points
sage: for i in xrange(3): points.append((random(), random()))
....:
sage: time R.lagrange_polynomial(points, neville=True, previous_row=p);
CPU times: user 0.09 s, sys: 0.00 s, total: 0.09 s
Wall time: 0.08 s
sage: p = R.lagrange_polynomial(points, neville=True, previous_row=p)
sage: # add another 100 points
sage: for i in xrange(100): points.append((random(), random()))
....:
sage: time R.lagrange_polynomial(points, neville=True, previous_row=p);
CPU times: user 4.62 s, sys: 0.00 s, total: 4.62 s
Wall time: 4.62 s
 }}}

Sage 4.0.1 Release Tour

Sage 4.0.1 was released on FIXME. For the official, comprehensive release note, please refer to FIXME. A nicely formatted version of this release tour can be found at FIXME. The following points are some of the foci of this release:

Algebra

  • Factoring rational functions (Soroosh Yazdani) -- New method factor() in the class FractionFieldElement of sage/rings/fraction_field_element.pyx to return the factorization of self over the base ring. Here's an example for working with this new method:

    sage: K.<x> = QQ["x"]
    sage: f = (x^3 + x) / (x-3)
    sage: f.factor()
    (x - 3)^-1 * x * (x^2 + 1)
  • Faster basis_matrix() for ambient modules (John Cremona) -- The speed-up can be up to 376x faster than previously. The following timing statistics were obtained using the machine sage.math:

    # BEFORE
    
    sage: K = FreeModule(ZZ, 2000)
    sage: %time I = K.basis_matrix()
    CPU times: user 292.74 s, sys: 20.11 s, total: 312.85 s
    Wall time: 312.90 s
    
    
    # AFTER
    
    sage: K = FreeModule(ZZ, 2000)
    sage: %time I = K.basis_matrix()
    CPU times: user 0.41 s, sys: 0.43 s, total: 0.84 s
    Wall time: 0.83 s
  • Optimize the construction of Lagrange interpolation polynomials (Minh Van Nguyen) -- Rewrite the method lagrange_polynomial() in the class PolynomialRing_field of sage/rings/polynomial/polynomial_ring.py for generating the n-th Lagrange interpolation polynomial. The method now provides two new options:

    • algorithm --- (default: divided_difference) If algorithm="divided_difference", then use the method of divided difference. If algorithm="neville", then use a variant of Neville's method to recursively generate the n-th Lagrange interpolation polynomial. This adaptation of Neville's method is more memory efficient than the original Neville's method, since the former doesn't generate the full Neville table resulting from Neville's recursive procedure. Instead the adaptation only keeps track of the current and previous rows of the said table.

    • previous_row --- (default: None) This is only relevant if used together with algorithm="neville". Here "previous row" refers to the last row in the Neville table that was obtained from a previous computation of an n-th Lagrange interpolation polynomial using Neville's method. If the last row is provided, then use a memory efficient variant of Neville's method to recursively generate a better interpolation polynomial from the results of previous computation.

    There's also the new method divided_difference() to compute the Newton divided-difference coefficients of the n-th Lagrange interpolation polynomial. The following are some timing statistics obtained using sage.math. When the results of previous computations are fed to lagrange_polynomial in order to produce better interpolation polynomials, we can gain an efficiency of up to 42%.

    # BEFORE
    
    # using the definition of Lagrange interpolation polynomial
    sage: R = PolynomialRing(QQ, 'x')
    sage: %timeit R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)])
    1000 loops, best of 3: 1.71 ms per loop
    sage: R = PolynomialRing(GF(2**3,'a'), 'x')
    sage: a = R.base_ring().gen()
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)])")
    625 loops, best of 3: 233 µs per loop
    
    # without using precomputed values to generate successively better interpolation polynomials
    
    sage: R = PolynomialRing(QQ, 'x')
    sage: timeit("R.lagrange_polynomial([(0,1),(2,2)])");
    625 loops, best of 3: 571 µs per loop
    sage: # add two more points
    sage: timeit("R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)])");
    125 loops, best of 3: 2.29 ms per loop
    sage: 
    sage: R = PolynomialRing(GF(2**3,'a'), 'x')
    sage: a = R.base_ring().gen()
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1)])")
    625 loops, best of 3: 76.1 µs per loop
    sage: # add another point
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)])")
    625 loops, best of 3: 229 µs per loop
    sage:
    sage: R = PolynomialRing(QQ, 'x')
    sage: points = [(random(), random()) for i in xrange(100)]
    sage: time R.lagrange_polynomial(points);
    CPU times: user 1.21 s, sys: 0.00 s, total: 1.21 s
    Wall time: 1.21 s
    sage: # add three more points
    sage: for i in xrange(3): points.append((random(), random()))
    ....: 
    sage: time R.lagrange_polynomial(points);
    CPU times: user 1.28 s, sys: 0.01 s, total: 1.29 s
    Wall time: 1.29 s
    sage: # add another 100 points
    sage: for i in xrange(100): points.append((random(), random()))
    ....: 
    sage: time R.lagrange_polynomial(points);
    CPU times: user 5.87 s, sys: 0.02 s, total: 5.89 s
    Wall time: 5.89 s
    
    
    # AFTER
    
    # using the method of divided-difference
    sage: R = PolynomialRing(QQ, 'x')
    sage: %timeit R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)])
    1000 loops, best of 3: 827 µs per loop
    sage: R = PolynomialRing(GF(2**3,'a'), 'x')
    sage: a = R.base_ring().gen()
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)])")
    625 loops, best of 3: 111 µs per loop
    
    # using precomputed values to generate successively better interpolation polynomials
    
    sage: R = PolynomialRing(QQ, 'x')
    sage: timeit("R.lagrange_polynomial([(0,1),(2,2)], neville=True)");
    625 loops, best of 3: 332 µs per loop
    sage: p = R.lagrange_polynomial([(0,1),(2,2)], neville=True);
    sage: # add two more points
    sage: timeit("R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)], neville=True, previous_row=p)");
    625 loops, best of 3: 1.41 ms per loop
    sage:
    sage: R = PolynomialRing(GF(2**3,'a'), 'x')
    sage: a = R.base_ring().gen()
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1)], neville=True)");
    625 loops, best of 3: 36.4 µs per loop
    sage: p = R.lagrange_polynomial([(a^2+a,a),(a,1)], neville=True);
    sage: # add another point
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)], neville=True, previous_row=p)");
    625 loops, best of 3: 131 µs per loop
    sage:
    sage: R = PolynomialRing(QQ, 'x')
    sage: points = [(random(), random()) for i in xrange(100)]
    sage: time R.lagrange_polynomial(points, neville=True);
    CPU times: user 1.26 s, sys: 0.00 s, total: 1.26 s
    Wall time: 1.26 s
    sage: p = R.lagrange_polynomial(points, neville=True);
    sage: # add three more points
    sage: for i in xrange(3): points.append((random(), random()))
    ....: 
    sage: time R.lagrange_polynomial(points, neville=True, previous_row=p);
    CPU times: user 0.09 s, sys: 0.00 s, total: 0.09 s
    Wall time: 0.08 s
    sage: p = R.lagrange_polynomial(points, neville=True, previous_row=p)
    sage: # add another 100 points
    sage: for i in xrange(100): points.append((random(), random()))
    ....: 
    sage: time R.lagrange_polynomial(points, neville=True, previous_row=p);
    CPU times: user 4.62 s, sys: 0.00 s, total: 4.62 s
    Wall time: 4.62 s

Algebraic Geometry

  • FIXME: summarize #5948

Basic Arithmetic

  • FIXME: summarize #5732

Calculus

  • FIXME: summarize #5404

Combinatorics

  • FIXME: summarize #6000
  • FIXME: summarize #6167
  • FIXME: summarize #6093
  • FIXME: summarize #6050
  • FIXME: summarize #5931
  • FIXME: summarize #5925

Commutative Algebra

  • FIXME: summarize #6120

Geometry

Graph Theory

Graphics

  • FIXME: summarize #6184
  • FIXME: summarize #5599

Group Theory

Interfaces

  • FIXME: summarize #6208

Linear Algebra

Miscellaneous

  • FIXME: summarize #5967
  • FIXME: summarize #5483
  • FIXME: summarize #6139

Modular Forms

  • FIXME: summarize #5995

Notebook

  • FIXME: summarize #4575
  • FIXME: summarize #5895

Number Theory

  • FIXME: summarize #133
  • FIXME: summarize #6021
  • FIXME: summarize #6206

Numerical

  • FIXME: summarize #5827

Packages

  • FIXME: summarize #5840
  • FIXME: summarize #6173
  • FIXME: summarize #5817
  • FIXME: summarize #6156
  • FIXME: summarize #6169
  • FIXME: summarize #6209
  • FIXME: summarize #6219

P-adics

Quadratic Forms

Symbolics

  • FIXME: summarize #6144
  • FIXME: summarize #6194

Topology

  • FIXME: summarize #6141

ReleaseTours/sage-4.0.1 (last edited 2019-11-14 21:01:15 by chapoton)