Numerical/examples.py at master · ravenwit/Numerical · GitHub

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from pybs.utils import treeType, number_of_trees_up_to_order
from pybs.unordered_tree import ButcherTree, ButcherEmptyTree, F, order, \
    number_of_children, density, symmetry, alpha
from pybs.combinations import Forest, differentiate as D, graft, treeCommutator, split, treeGenerator
from pybs.series import BseriesRule, hf_composition, modifiedEquation

print('The first tree is: ButcherTree() =', ButcherTree())
print('And the empty tree is: ButcherEmptyTree() =', ButcherEmptyTree())
print('And since F(ButcherEmptyTree()) =', F(ButcherEmptyTree()), ', we have D(ButcherEmptyTree) =', str(D(ButcherEmptyTree())) + '. Note: The result is a "LinearCombination" containing one "' + str(ButcherTree.basetree()) + '".')
print('These two first trees can be accessed through ButcherTree.emptytree() =', ButcherTree.emptytree(), 'and ButcherTree.basetree() =', str(ButcherTree.basetree()) + '.')
print('Some methods for constructing new trees are:')
print('    1. by specifying the forest of child trees to the constructor.')
tree = ButcherTree(Forest([ButcherTree.basetree(), ButcherTree.basetree()]))
print('    Ex.: tree = ButcherTree(Forest([ButcherTree.basetree(), ButcherTree.basetree()])) = ', tree)
print('    2. by grafting one tree onto another:')
print('    Ex.: graft(tree, ButcherTree.basetree()) = ', graft(tree, ButcherTree.basetree()))
print('    3. by taking the derivative of existing trees.')
print('    Ex.: D(ButcherTree.basetree()) =', D(ButcherTree.basetree()))

print('It is also possible to take the derivative of all the trees in a LinearCOmbination. Starting at the empty tree,')
print('the n-th derivative is a LinearCombination of all trees of order n, complete with alpha as the multiplicity:')
trees_of_order_1 = D(ButcherTree.emptytree())
print('trees_of_order_1 = D(ButcherTree.emptytree()) =', trees_of_order_1)
trees_of_order_2 = D(trees_of_order_1)
print('trees_of_order_2 = D(trees_of_order_1) =', trees_of_order_2)
trees_of_order_3 = D(trees_of_order_2)
print('trees_of_order_3 = D(trees_of_order_2) =', trees_of_order_3)
trees_of_order_4 = D(trees_of_order_3)
print('trees_of_order_4 = D(trees_of_order_3) =', trees_of_order_4)
trees_of_order_5 = D(trees_of_order_4)
print('trees_of_order_5 = D(trees_of_order_4) =', trees_of_order_5)
print('The number of trees of a given order can be calculated')
print('without constructing all the trees: number_of_trees_of_order(30) = ', number_of_trees_of_order(30))
print("")
print('Some functions on trees:')
tmp = ButcherTree(Forest([ButcherTree.basetree()]))
t = ButcherTree(Forest([tmp, ButcherTree.basetree()])); del tmp
print('let t =', str(t) + ', then:')
print('order(t) = ', order(t))
print('number_of_children(t) = ', number_of_children(t))
print('density(t) = ', density(t))
print('symmetry(t) = ', symmetry(t))
print('alpha(t) =', alpha(t))
print('F(t) = "' + F(t) + '" (A string)')
print('t.multiplicities() =', t.multiplicities(),  '(A, list)')
print("")
print('Tree commutator: treeCommutator(t, ButcherTree.basetree()) = ', treeCommutator(t, ButcherTree.basetree()))
print('Splitting: split(tree) = ', split(tree))
print("")
print('B-series:')
a = BseriesRule('exact')
print("Let a = BseriesRule('exact')")
b = hf_composition(a)
print("Then b = hf_composition(a) gives:")
b = hf_composition(a)
tmp = treeGenerator()
tmp_tree = tmp.next()
print(tmp_tree, ', b = ', b(tmp_tree))
tmp_tree = tmp.next()
print(tmp_tree, ', b = ', b(tmp_tree))
tmp_tree = tmp.next()
print(tmp_tree, ', b = ', b(tmp_tree))
tmp_tree = tmp.next()
print(tmp_tree, ', b = ', b(tmp_tree))

c = modifiedEquation(a)
print('If c = modifiedEquation(a), one gets the B series corresponding to the original equation:')
tmp = treeGenerator()
tmp_tree = tmp.next()
print(tmp_tree, ', c = ', c(tmp_tree))
tmp_tree = tmp.next()
print(tmp_tree, ', c = ', c(tmp_tree))
tmp_tree = tmp.next()
print(tmp_tree, ', c = ', c(tmp_tree))
tmp_tree = tmp.next()
print(tmp_tree, ', c = ', c(tmp_tree), '(c is zero for all other trees...)')
print("")
from pybs.rungekutta.methods import RKmidpoint # A small collection of Butcher tableaus.
print('And last, but not least, finding the order of an RK method.')
print('Ex.: The midpoint rule:')
print('RKmidpoint.printMe():')
RKmidpoint.printMe()
print('RKmidpoint.order =', RKmidpoint.order)