Trac #32324: Lazy Taylor Series

We implement (multivariate) Taylor series.

Note that there is no easy well-defined notion of multivariate Laurent
series, which is why we create a new class.

URL: https://trac.sagemath.org/32324
Reported by: mantepse
Ticket author(s): Martin Rubey
Reviewer(s): Travis Scrimshaw

src/sage/data_structures/stream.py

@@ -4,7 +4,7 @@
 This module provides lazy implementations of basic operators on
 streams. The classes implemented in this module can be used to build
 up more complex streams for different kinds of series (Laurent,
-Dirichlet, etc).
+Dirichlet, etc.).
 
 EXAMPLES:
 
@@ -85,6 +85,8 @@
 
 # ****************************************************************************
 #       Copyright (C) 2019 Kwankyu Lee <ekwankyu@gmail.com>
+#                     2022 Martin Rubey <martin.rubey at tuwien.ac.at>
+#                     2022 Travis Scrimshaw <tcscrims at gmail.com>
 #
 # This program is free software: you can redistribute it and/or modify
 # it under the terms of the GNU General Public License as published by
@@ -96,6 +98,7 @@
 from sage.rings.integer_ring import ZZ
 from sage.rings.infinity import infinity
 from sage.arith.misc import divisors
+from sage.combinat.integer_vector_weighted import iterator_fast as wt_int_vec_iter
 
 class Stream():
     """
@@ -505,7 +508,7 @@ class Stream_exact(Stream):
     """
     def __init__(self, initial_coefficients, is_sparse, constant=None, degree=None, order=None):
         """
-        Initialize a series that is known to be eventually geometric.
+        Initialize a stream with eventually constant coefficients.
 
         TESTS::
 
@@ -837,6 +840,7 @@ def __init__(self, is_sparse, approximate_order):
             sage: TestSuite(C).run(skip="_test_pickling")
         """
         self._target = None
+        assert approximate_order is not None, "calling Stream_uninitialized with None as approximate order"
         super().__init__(is_sparse, approximate_order)
 
     def get_coefficient(self, n):
@@ -1023,7 +1027,7 @@ def __eq__(self, other):
 
         INPUT:
 
-        - ``other`` -- a stream of coefficients
+        - ``other`` -- a :class:`Stream` of coefficients
 
         EXAMPLES::
 
@@ -1086,7 +1090,7 @@ def __eq__(self, other):
 
         INPUT:
 
-        - ``other`` -- a stream of coefficients
+        - ``other`` -- a :class:`Stream` of coefficients
 
         EXAMPLES::
 
@@ -1211,8 +1215,8 @@ class Stream_add(Stream_binaryCommutative):
 
     INPUT:
 
-    - ``left`` -- stream of coefficients on the left side of the operator
-    - ``right`` -- stream of coefficients on the right side of the operator
+    - ``left`` -- :class:`Stream` of coefficients on the left side of the operator
+    - ``right`` -- :class:`Stream` of coefficients on the right side of the operator
 
     EXAMPLES::
 
@@ -1271,8 +1275,8 @@ class Stream_sub(Stream_binary):
 
     INPUT:
 
-    - ``left`` -- stream of coefficients on the left side of the operator
-    - ``right`` -- stream of coefficients on the right side of the operator
+    - ``left`` -- :class:`Stream` of coefficients on the left side of the operator
+    - ``right`` -- :class:`Stream` of coefficients on the right side of the operator
 
     EXAMPLES::
 
@@ -1336,8 +1340,8 @@ class Stream_cauchy_mul(Stream_binary):
 
     INPUT:
 
-    - ``left`` -- stream of coefficients on the left side of the operator
-    - ``right`` -- stream of coefficients on the right side of the operator
+    - ``left`` -- :class:`Stream` of coefficients on the left side of the operator
+    - ``right`` -- :class:`Stream` of coefficients on the right side of the operator
 
     EXAMPLES::
 
@@ -1422,8 +1426,8 @@ class Stream_dirichlet_convolve(Stream_binary):
 
     INPUT:
 
-    - ``left`` -- stream of coefficients on the left side of the operator
-    - ``right`` -- stream of coefficients on the right side of the operator
+    - ``left`` -- :class:`Stream` of coefficients on the left side of the operator
+    - ``right`` -- :class:`Stream` of coefficients on the right side of the operator
 
     The coefficient of `n^{-s}` in the convolution of `l` and `r`
     equals `\sum_{k | n} l_k r_{n/k}`.
@@ -1598,7 +1602,7 @@ def __init__(self, f, g):
             sage: g = Stream_function(lambda n: n^2, ZZ, True, 1)
             sage: h = Stream_cauchy_compose(f, g)
         """
-        assert g._approximate_order > 0
+        #assert g._approximate_order > 0
         self._fv = f._approximate_order
         self._gv = g._approximate_order
         if self._fv < 0:
@@ -1643,6 +1647,132 @@ def get_coefficient(self, n):
         return ret + sum(self._left[i] * self._pos_powers[i][n] for i in range(1, n // self._gv+1))
 
 
+class Stream_plethysm(Stream_binary):
+    r"""
+    Return the plethysm of ``f`` composed by ``g``.
+
+    This is the plethysm `f \circ g = f(g)` when `g` is an element of
+    the ring of symmetric functions.
+
+    INPUT:
+
+    - ``f`` -- a :class:`Stream`
+    - ``g`` -- a :class:`Stream` with positive order
+    - ``p`` -- the powersum symmetric functions
+
+    EXAMPLES::
+
+        sage: from sage.data_structures.stream import Stream_function,  Stream_plethysm
+        sage: s = SymmetricFunctions(QQ).s()
+        sage: p = SymmetricFunctions(QQ).p()
+        sage: f = Stream_function(lambda n: s[n], s, True, 1)
+        sage: g = Stream_function(lambda n: s[[1]*n], s, True, 1)
+        sage: h = Stream_plethysm(f, g, p)
+        sage: [s(h[i]) for i in range(5)]
+        [0,
+         s[1],
+         s[1, 1] + s[2],
+         2*s[1, 1, 1] + s[2, 1] + s[3],
+         3*s[1, 1, 1, 1] + 2*s[2, 1, 1] + s[2, 2] + s[3, 1] + s[4]]
+        sage: u = Stream_plethysm(g, f, p)
+        sage: [s(u[i]) for i in range(5)]
+        [0,
+         s[1],
+         s[1, 1] + s[2],
+         s[1, 1, 1] + s[2, 1] + 2*s[3],
+         s[1, 1, 1, 1] + s[2, 1, 1] + 3*s[3, 1] + 2*s[4]]
+    """
+    def __init__(self, f, g, p):
+        r"""
+        Initialize ``self``.
+
+        TESTS::
+
+            sage: from sage.data_structures.stream import Stream_function, Stream_plethysm
+            sage: s = SymmetricFunctions(QQ).s()
+            sage: p = SymmetricFunctions(QQ).p()
+            sage: f = Stream_function(lambda n: s[n], s, True, 1)
+            sage: g = Stream_function(lambda n: s[n-1,1], s, True, 2)
+            sage: h = Stream_plethysm(f, g, p)
+        """
+        #assert g._approximate_order > 0
+        self._fv = f._approximate_order
+        self._gv = g._approximate_order
+        self._p = p
+        val = self._fv * self._gv
+        super().__init__(f, g, f._is_sparse, val)
+
+    def get_coefficient(self, n):
+        r"""
+        Return the ``n``-th coefficient of ``self``.
+
+        INPUT:
+
+        - ``n`` -- integer; the degree for the coefficient
+
+        EXAMPLES::
+
+            sage: from sage.data_structures.stream import Stream_function, Stream_plethysm
+            sage: s = SymmetricFunctions(QQ).s()
+            sage: p = SymmetricFunctions(QQ).p()
+            sage: f = Stream_function(lambda n: s[n], s, True, 1)
+            sage: g = Stream_function(lambda n: s[[1]*n], s, True, 1)
+            sage: h = Stream_plethysm(f, g, p)
+            sage: s(h.get_coefficient(5))
+            4*s[1, 1, 1, 1, 1] + 4*s[2, 1, 1, 1] + 2*s[2, 2, 1] + 2*s[3, 1, 1] + s[3, 2] + s[4, 1] + s[5]
+            sage: [s(h.get_coefficient(i)) for i in range(6)]
+            [0,
+             s[1],
+             s[1, 1] + s[2],
+             2*s[1, 1, 1] + s[2, 1] + s[3],
+             3*s[1, 1, 1, 1] + 2*s[2, 1, 1] + s[2, 2] + s[3, 1] + s[4],
+             4*s[1, 1, 1, 1, 1] + 4*s[2, 1, 1, 1] + 2*s[2, 2, 1] + 2*s[3, 1, 1] + s[3, 2] + s[4, 1] + s[5]]
+        """
+        if not n: # special case of 0
+            return self._left[0]
+
+        # We assume n > 0
+        p = self._p
+        ret = p.zero()
+        for k in range(n+1):
+            temp = p(self._left[k])
+            for la, c in temp:
+                inner = self._compute_product(n, la, c)
+                if inner is not None:
+                    ret += inner
+        return ret
+
+    def _compute_product(self, n, la, c):
+        """
+        Compute the product ``c * p[la](self._right)`` in degree ``n``.
+
+        EXAMPLES::
+
+            sage: from sage.data_structures.stream import Stream_plethysm, Stream_exact, Stream_function
+            sage: s = SymmetricFunctions(QQ).s()
+            sage: p = SymmetricFunctions(QQ).p()
+            sage: f = Stream_function(lambda n: s[n], s, True, 1)
+            sage: g = Stream_exact([s[2], s[3]], False, 0, 4, 2)
+            sage: h = Stream_plethysm(f, g, p)
+            sage: ret = h._compute_product(7, [2, 1], 1); ret
+            1/12*p[2, 2, 1, 1, 1] + 1/4*p[2, 2, 2, 1] + 1/6*p[3, 2, 2]
+             + 1/12*p[4, 1, 1, 1] + 1/4*p[4, 2, 1] + 1/6*p[4, 3]
+            sage: ret == p[2,1](s[2] + s[3]).homogeneous_component(7)
+            True
+        """
+        p = self._p
+        ret = p.zero()
+        for mu in wt_int_vec_iter(n, la):
+            temp = c
+            for i, j in zip(la, mu):
+                gs = self._right[j]
+                if not gs:
+                    temp = p.zero()
+                    break
+                temp *= p[i](gs)
+            ret += temp
+        return ret
+
 #####################################################################
 # Unary operations
 
@@ -2194,3 +2324,4 @@ def is_nonzero(self):
             True
         """
         return self._series.is_nonzero()
+

src/sage/rings/all.py

@@ -146,7 +146,8 @@
 lazy_import('sage.rings.laurent_series_ring_element', 'LaurentSeries', deprecation=33602)
 
 # Lazy Laurent series ring
-lazy_import('sage.rings.lazy_series_ring', ['LazyLaurentSeriesRing', 'LazyDirichletSeriesRing'])
+lazy_import('sage.rings.lazy_series_ring', ['LazyLaurentSeriesRing', 'LazyTaylorSeriesRing',
+                                            'LazySymmetricFunctions', 'LazyDirichletSeriesRing'])
 
 # Tate algebras
 from .tate_algebra import TateAlgebra