6,235 research outputs found
Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems
We consider solution operators of linear ordinary boundary problems with "too
many" boundary conditions, which are not always solvable. These generalized
Green's operators are a certain kind of generalized inverses of differential
operators. We answer the question when the product of two generalized Green's
operators is again a generalized Green's operator for the product of the
corresponding differential operators and which boundary problem it solves.
Moreover, we show that---provided a factorization of the underlying
differential operator---a generalized boundary problem can be factored into
lower order problems corresponding to a factorization of the respective Green's
operators. We illustrate our results by examples using the Maple package
IntDiffOp, where the presented algorithms are implemented.Comment: 19 page
Holonomic Gradient Descent and its Application to Fisher-Bingham Integral
We give a new algorithm to find local maximum and minimum of a holonomic
function and apply it for the Fisher-Bingham integral on the sphere ,
which is used in the directional statistics. The method utilizes the theory and
algorithms of holonomic systems.Comment: 23 pages, 1 figur
Can Computer Algebra be Liberated from its Algebraic Yoke ?
So far, the scope of computer algebra has been needlessly restricted to exact
algebraic methods. Its possible extension to approximate analytical methods is
discussed. The entangled roles of functional analysis and symbolic programming,
especially the functional and transformational paradigms, are put forward. In
the future, algebraic algorithms could constitute the core of extended symbolic
manipulation systems including primitives for symbolic approximations.Comment: 8 pages, 2-column presentation, 2 figure
Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control
Variational integrators are well-suited for simulation of mechanical systems
because they preserve mechanical quantities about a system such as momentum, or
its change if external forcing is involved, and holonomic constraints. While
they are not energy-preserving they do exhibit long-time stable energy
behavior. However, variational integrators often simulate mechanical system
dynamics by solving an implicit difference equation at each time step, one that
is moreover expressed purely in terms of configurations at different time
steps. This paper formulates the first- and second-order linearizations of a
variational integrator in a manner that is amenable to control analysis and
synthesis, creating a bridge between existing analysis and optimal control
tools for discrete dynamic systems and variational integrators for mechanical
systems in generalized coordinates with forcing and holonomic constraints. The
forced pendulum is used to illustrate the technique. A second example solves
the discrete LQR problem to find a locally stabilizing controller for a 40 DOF
system with 6 constraints.Comment: 13 page
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