Re: Unstable Simultaneous Linear Equations!

On Apr 4, 11:04 pm, The Phantom <phan...@xxxxxxx> wrote:
On Fri, 4 Apr 2008 11:30:28 -0700 (PDT), monir <mon...@xxxxxxxxxxxx> wrote:
On Apr 3, 11:09 pm, The Phantom <phan...@xxxxxxx> wrote:
On Thu, 3 Apr 2008 18:14:51 -0700 (PDT), monir <mon...@xxxxxxxxxxxx> wrote:

<SNIP>
Can you show some of the points in your analytical function
that are outside the interval P1 < x < P1?

You didn't answer this question and I think there may be some mis-communication
about what you want because of this very thing.





I assumed that as you approach P2
from the left, you would want to be moving up and to the right.

The Phantom;

"I think what Dave is talking about is something that I also wondered about."
I've recalculated the elliptic function around the vertical tangent to
the right and leading to point P2, with x-interval ~ 2.E-7

Given: (exact 7 significant digits)
..........x.........y........d=(dy/dx)
P1:: -0.0266255  0.9981449   -0.1308565
P2:: -0.0248588  0.9847863  132.5335679

pnt.......X.......Y results...(dY/dX)results
1    -0.0248585   0.9860975
2    -0.0248583   0.9860610
3    -0.0248580   0.9860221
4    -0.0248578   0.9859804
5    -0.0248575   0.9859350
6    -0.0248573   0.9858848
7    -0.0248570   0.9858278
8    -0.0248568   0.9857601
9    -0.0248565   0.9856720
10   -0.0248564   0.9856096
11   -0.0248563   0.9854591  vertical tangent
12  -0.0248563   0.9854591  vertical tangent
13  -0.0248564   0.9853086
14  -0.0248565   0.9852463
15  -0.0248568   0.9851582
16  -0.0248570   0.9850905
17  -0.0248573   0.9850335
18  -0.0248575   0.9849833
19  -0.0248578   0.9849379
20  -0.0248580   0.9848961
21  -0.0248583   0.9848573
22  -0.0248585   0.9848208
23  -0.0248588   0.9847863  132.5335679

Please identify which point has a "cusp" ?? You may refer to pnt #
instead of coordinates.

These points are on the very right end of the ellipse and there is no cusp
associated with this curve BY ITSELF.  



"... and the interpolating curve will have to have a "minimum" as the cubic does."
Please identify where is the "minimum" ??

The top half of the elliptic curve which connects P1 to P2 doesn't have a
minimum of the sort I was talking about.



So far, I've tested the elliptic procedure for about 60 different
pairs, with different scenarios scen1, scen2, scen3, scen4, and with
different signs and magnitudes of given slopes d1 and d2, and the
procedure appears to be working fine!

Any comments or thoughts ??  Thank you.
Monir

I have assumed all along that there more points, or another "analytical" curve,
to the left of P1 and to the right of P2, and that the curve you are attempting
to derive should "fill in the gap", meeting the points (or curve) outside the
interval P1 < x < P1, matching in position and slope at P1 and P2.

Imagine that you start at P1 and move on some curve toward P2. When you arrive
at P2 I assume that you want to be moving on a slope of ~132, upward and to the
right, NOT downward and to the left.

You haven't said which one of these you want, but I assumed you want to be
heading northeast, not southwest.

If you use the ellipse as you have shown, when you reach P2, you will be heading
southwest; the slope will be ~132, but you will be heading in a direction that
takes you back in the the x-interval BETWEEN P1 and P2, rather than out to the
right.  You will have reversed your travel in the x-coordinate.

As I say, I have assumed that you want to be going northeast when you reach P2.

If you DO want to be going northeast, then if you travel from P1 to P2 via the
elliptic curve, you must reverse direction when you reach P2 in order to
continue out of the interval in a northeasterly direction.

Have a look at this graphic:

http://img2.freeimagehosting.net/image.php?13225904c8.gif

The top graph shows the red elliptic curve going from P1 to P2, but when you get
to P2, if you want to continue to leave the P1-P2 interval in a northearterly
direction, you must reverse direction and follow the black line, which is just a
line with a slope of ~132.  Where the elliptic curve and the black line meet,
there is a cusp.  Your elliptic curve doesn't have a cusp by itself, but the
joining of the elliptic curve and the continuation line does.

Perhaps I've misunderstood, and you don't care about a continuation leaving the
P1-P2 interval to the northeast.  If so, then perhaps the elliptic curve meets
your needs.

If you look at the bottom graph, a repeat of an earlier one, you see the blue
cubic curve has a local minimum between P1 and P2.  What Dave was saying is
this;  If you want a curve that moves from P1 to P2, leaving P1 with a slope of
-0.1308565 and then approaching P2 with a slope of 132.5335679, you have two
choices of how to approach P2.  You can approach with a northeasterly direction
of travel or with a southwesterly direction of travel.  If you approach heading
to the northeast, you must be approaching from the left and below.  Since the
y-coordinate of P2 is less than that of P1, you must first go below P2's y
coordinate in order to then approach while travelling northeasterly.  This means
that there MUST be a local minimum in the cubic or any other curve that would
approach in the manner I described.  But your elliptic curve has no minimum OF
THAT SORT.  It does have a minimum where it meets P2, but it's the end of the
elliptic curve, not between P1 and P2 as is the local minimum of the blue cubic
curve.

Perhaps all of this has no relevance to what you want to do; perhaps what
happens outside of the interval P1 < x < P2 doesn't matter to you.  But I
thought you wanted your curve to be a continuation in position and slope of the
data outside the interval.

I didn't mean (and I don't think Dave did either) that the cusp and minimum were
part of your elliptic curve.  The cusp is due to the connection of the elliptic
curve with a path leading out of the interval to the northeast.

However, it's certainly possible to have a path leading away from P2 to the
southwest as a continuation of your elliptic curve after it reaches P2.  It
would be an extension of the black line in the southwesterly direction.  In that
case there will be no cusp, but I didn't think you would want to be following a
path that would lead your x-coordinate back into the interval between P1 and P2.- Hide quoted text -

- Show quoted text -

Rodger;

Tremendous effort! Greatly appreciated and acknowledged!

1) >"I assumed that as you approach P2 from the left, you would want
to be moving up and to the right."
"Imagine that you start at P1 and move on some curve toward P2. When you arrive at P2 I assume that you want to be moving on a slope of ~132, upward and to the right, NOT downward and to the left."
"As I say, I have assumed that you want to be going northeast when you reach P2."

Why did you assume that ??

2) In an earlier reply, I summarized the problem and the conditions/
requirements of the sought function/curve joining P1(x1,y1) and
P2(x2,y2) as follows:
General:
x1 < x2 < 0
y1 > y2 > 0
sign of d1::(dy/dx) at P1 and d2::(dy/dx) at P2 could be any of the 4
+/- combinations.
Desired Solution:
a. satisfies the position and slope at the two endpoints;
b- numerically stable and does not require extra precision;
c. exact, smooth, continuous and well-defined;
d. interpolated y-value is bounded by y2 and y1 values; and
e. presents no extremum or inflection points within the range-of-
interest.
What happens to the interpolation curve leaving P2 is not and should
not be a concern.

3) >"I didn't mean (and I don't think Dave did either) that the cusp
and minimum were part of your elliptic curve. The cusp is due to the
connection of the elliptic curve with a path leading out of the
interval to the northeast."

I just wonder: Are you using a different definition of "cusp" ?? It
usually refers to a spike in the numerical values! In aeronautical
engineering, we say the trailing edge is "cusped" if the angle between
the upper and lower surfaces of the aerofoil is zero, a definition
which is analogous/consistent with the NA definition.

4) If you still see a "cusp due to the connection of the elliptic
curve with the path leading out ", please provide the corresponding x
value and the value of the "cusp".

5) I'm sorry but I don't see how the characteristics of the
"analytical" curve approaching P1 from the left and the curve leaving
P2 to the left (for the worked example) affect the elliptic solution
or produce a cusp.
Let us assume for the sake of discussion that both "analytical" curves
are straight-line segments with slope d1 and d2 respectively.

6) Should there be a better function/curve that satisfies a-d above, I
would be anxious to try it.

Any thoughts ?? Thanks again for your help and interest in the
problem. I find the discussion very useful!

Monir
.

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