Open 2b5de62a-05ae-46d3-a99b-c6405c33a11e opened 12 years ago
In the above example there is supposed to be a DifferentialAlgebra.pyx in this branch of sage:
see the examples at the very beginning which has the line:
from sage.libs.blad.DifferentialAlgebra import DifferentialRing, RegularDifferentialChain, BaseFieldExtension
Where is this now?
Replying to @tdupu:
In the above example there is supposed to be a DifferentialAlgebra.pyx in this branch of sage:
see the examples at the very beginning which has the line:
from sage.libs.blad.DifferentialAlgebra import DifferentialRing, RegularDifferentialChain, BaseFieldExtension
Where is this now?
DifferentialRing
is defined in src/sage/calculus/DifferentialAlgebra.pyx
You also need to install the BLAD and BMI libraries.
If you checkout the public/optional_spkg/differential_algebra-13268
branch and run git diff 9.0.beta4 --stat
, that should give you a good idea of where things are in this branch.
This ticket proposes the inclusion in SAGE of a new package, dedicated to differential algebra.
Introduction
The
DifferentialAlgebra
Sage package is an analogue of the MAPLE 14DifferentialAlgebra
package. The underlying theory is the differential algebra of Ritt and Kolchin. Its main tool is a simplifier for systems of polynomial differential equations, ordinary or with partial derivatives, calledRosenfeldGroebner
. It is related to the differential elimination theory. This simplifier decomposes the radical differential ideal I generated by an input system, as an intersection of radical differential ideals presented by regular differential chains (a slight generalization of Ritt characteristic sets). The output permits to test membership in the differential ideal I.Further developments
Software
The package is written in Cython. The computations are performed by the BLAD libraries (C libraries, 60000 lines, LGPL license). The interface between Sage and BLAD is handled by the BMI library (C library, 10000 lines, LGPL license).
Upstream: BLAD BMI
Installation
After building Sage, install the experimental packages
blad
andbmi
, then runmake
ormake build
again, i.e:An example
Borrowed from
DifferentialAlgebra.pyx
, to motivate (hopefully) reviewers.This example shows how to build the Henri Michaelis Menten formula by differential elimination. One considers a chemical reaction system describing the enzymatic reaction:
A substrate S is transformed into a product P, in the presence of an enzyme E. An intermediate complex ES is formed.
The main assumption is that k(1), k(-1) >> k(2) i.e. that the revertible reaction is much faster than the last one. One performs a quasi-steady state approximation by considering the following differential-algebraic system (it comes from the mass-action law kinetics, replacing the contribution of the fast reactions by an unknown function F_1(t), on the algebraic variety where the fast reaction would equilibrate if they were alone).
Differential elimination permits to simplify this DAE. To avoid discussing the possible vanishing of
params
, one moves them to the base field of the equations.The Rosenfeld-Groebner algorithm considers three cases. The two last ones are degenerate cases.
The sought equation, below, is not yet the Henri-Michaelis-Menten formula. This is expected, since some minor hypotheses have not yet been taken into account
Let us take them into account. First create two new constants. Put them among
params
, together with initial values.There are relations among the parameters: initial values supposed to be zero, and equations meant to rename constants.
Coming computations will be performed over a base field defined by generators and relations
Extend the DAE with linear conservation laws. They could have been computed from the stoichimetry matrix of the chemical system.
Simplify again. Only one case is left.
To get the traditional Henri-Michaelis-Menten formula, one still needs to neglect the term K*E(0)
One can also get it by computing the right hand side of the equation which gives the evolution of the product P
CC: @sagetrac-sage-combinat @sagetrac-boulier @burcin @kcrisman @BrentBaccala
Component: packages: optional
Keywords: package, differential algebra, elimination theory
Author: Nicolas M. Thiéry, François Boulier, Charles Bouillaguet
Branch/Commit: public/optional_spkg/differential_algebra-13268 @
81aef63
Reviewer: Charles Bouillaguet, Karl-Dieter Crisman
Issue created by migration from https://trac.sagemath.org/ticket/13268