MHB Reversing recurrence relationships

AI Thread Summary
Reversing recurrence relationships can be a valuable technique, especially when boundary conditions are inaccurate or unknown. Evaluating equations like the Legendre polynomial can be approached both forwards and backwards, with the forward direction solving for Pl+1 and the reverse for Pl-1. The arithmetic involved in manipulating these equations is straightforward, but clarity on the implications of each direction is essential. Numeric methods with clear step points allow for easy evaluation in both directions, while those with variable step sizes can also be reversed by adjusting the step. Overall, the discussion confirms that reversing relationships is valid and effective, provided the methods are understood correctly.
ognik
Messages
626
Reaction score
2
A couple of times I have come across the suggestion that numerically evaluating a recursive relation in reverse can be a valuable approach. I can see this where, for example, the boundary conditions at one 'end' are inaccurate or undiscoverable. However, while the arithmetic of manipulating such equations seems simple, I wonder if I am missing something?
One example is a Legendre polynomial, given by (l+1)Pl+1 + lPl - (2l+1)xPl=0
Should I evaluate this in the 'forward' direction, by solving for Pl+1, and in the reverse direction by solving for Pl-1? I am also struggling for some intuition as to what the difference(s) may be?
 
Mathematics news on Phys.org
After writing a few programs to compare, I am satisfied that when a numeric method has clear '3 points', then I can evaluate this in the 'forward' direction, by solving for Pl+1, and in the reverse direction by solving for Pl-1. In some other methods where steps are only visible as +(some step size, like h), then we can 'reverse' direction by using -h. This was all intuitively obvious, I just wanted confirmation I wasn't missing anything else ...
 

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Similar threads

Random Walks in 2D: Recurrence of (a), (b), (c)
Replies
6
Views
4K
MHB I'm Confused's question on Yahoo Answers involving a recurrence relation
Replies
2
Views
2K
Trying to solve any equation ever with Recurrences
Replies
10
Views
3K
I The relationship between reversible and irreversible processes
Replies
5
Views
2K
Mathematica Numerically solve recurrence relation
Replies
5
Views
2K
Legendre polynomials in the reverse direction
Replies
2
Views
2K
MHB Finding particular solution to recurrence relation
Replies
4
Views
4K
Understanding the Legendre Recurrence Relation for Generating Functions
Replies
1
Views
2K
MHB Solving for specific variable when solving a recurrence equation
Replies
1
Views
3K
Electrostatic potential of a split sphere, using the laplacian for spherical coord
Replies
1
Views
4K

Hot Threads

Recent Insights

Back
Top