I Who is Right: Backward Difference Table or Forward Difference Table?

[momentum]

<Moderator's note: Please upload images for otherwise the links might get broken.>

Here is the backward difference table given in my book

https://prnt.sc/n2965l

But it seems this table is wrong. I have marked my comment in the screen.

Can you please tell who is right ?

 

[jambaugh]

Yes you are correct they made an error.

 

[Ray Vickson]

<Moderator's note: Please upload images for otherwise the links might get broken.>

Here is the backward difference table given in my book

https://prnt.sc/n2965lView attachment 240773
But it seems this table is wrong. I have marked my comment in the screen.

Can you please tell who is right ?

(1) Yes, you are right.
(2) Isn't that a forward-difference table?

 

[jambaugh]

Whether it is a forward or backward difference table depends on how you allign entries. Since the table has the differences half way between the terms they subtract you can view them as forward differences (next row's entry minus this row's) by assuming the rows slant downward to the left. Letting the rows slant upward allows you to call them backward differences (this row's entry minus the previous row's).

 

[pbuk]

Whether it is a forward or backward difference table depends on how you align entries.

And on how you label them - the columns here are labelled $\nabla, \nabla^2 ...$ which by convention refers to backward differences i.e $x_n - x_{n-1}$ (forward differences $\Delta$ being $x_{n+1} - x_n$).

 

[jambaugh]

I ran into this with my Survey of Calc students this semester. I taught them a bit of discrete calculus because it makes things like the FTC rather obvious. But many are Econ students used to taking backward differences when I insist (so certain facts line up) that we work primarily with forward differences. So I don't mark their completed tables wrong but bless them out a bit on the comments when I return work. (Mainly about knowing which is which and being sure to follow the given instructions.)

Easy FTC proof: A sum of successive forward differences cancel all but the last minus first term. Multipy by h/h and the sum becomes a Riemann sum and the differences become difference quotients. Take the simultaneous limit and you have a definite integral of a derivative equals difference in end values.

 

[Ray Vickson]

And on how you label them - the columns here are labelled $\nabla, \nabla^2 ...$ which by convention refers to backward differences i.e $x_n - x_{n-1}$ (forward differences $\Delta$ being $x_{n+1} - x_n$).

Ok, I failed to look at the column labels---that is, I thought they were $\Delta$, etc.

However, to me the table looks weird; over many decades of viewing tables, I have never seen tables that you read from bottom to top; reading from top to bottom seems to be almost a universal convention (albeit undocumented), rather like that of reading from left to right in Western languages.