Re: Unstable Simultaneous Linear Equations!

On Sat, 5 Apr 2008 08:54:15 -0700 (PDT), monir <monirg@xxxxxxxxxxxx> wrote:

<SNIP>
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 ??

In an early post you said:

"I've a tiny "gap" in the analytical derivation between two points P1 and P2."

but then you say that you want to find a curve that has the proper derivative AT
the points P1 and P2. This makes it sound to me like the "gap" IS the entire
interval between P1 and P2. If the ends of the gap were not AT P1 and P2, you
would care about the derivative at points internal to P1 and P2 where the gap
begins and ends, not at exactly P1 and P2. And if the entire interval between
P1 and P2 IS, in fact, the "gap", then it would seem reasonable to assume that
there is something OUTSIDE the gap, otherwise it wouldn't be a "gap". The very
word "gap" implies something outside the "gap".

It would be EXTREMELY helpful if you would show, in a graph, the analytical
derivation you referred to, the "gap" in that derivation, and show P1 and P2 on
the plot.


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.

"What happens to the interpolation curve leaving P2 is not and should not be a
concern."

This point has not been clear to me all along. I explain above why I assumed
that there was something "outside" the P1-P2 interval, for which you were trying
create a continuation inside the P1-P2 interval.


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" ??

From my Dictionary of Mathematics:

"Cusp, A double point at which the two tangents to the curve are coincident. A
cusp of the first kind is a cusp in which there is a branch of the curve on EACH
side of the double tangent in the neighborhood of the point of tangency. A cusp
of the second kind is a cusp for which the two branches of the curve lie on the
SAME side of the tangent in the neighborhood of the point of tangency"

In this case the two tangents to the curve (the "curve" being the combination of
the elliptic curve, and the black continuation line) are the tangent to the
elliptic curve and the tangent to whatever continuation curve there may be. If
the continuation curve has the same slope at P2 as the elliptic curve, then they
form a cusp at P2 IF the continuation leaves P2 in a northeasterly direction.
Which kind of cusp depends on whether the continuation curve is concave up or
concave down. If the continuation curve is a straight line, then I guess it's a
cusp of the 1 and 1/2 kind. :-)

The dictionary gives an example: The semicubical parabola, y^2 = x^3, has a
cusp at the origin.

It >usually refers to a spike in the numerical values!

I've been using it not to refer to a spike in numerical values, but in the
characteristic of a curve at a point, in accordance with the mathematical
definition given above.

Also, see:

http://en.wikipedia.org/wiki/Cusp_%28singularity%29

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.

Have a look at the graphs in:

http://img2.freeimagehosting.net/image.php?04b4672dfb.gif

The upper graph shows your elliptic curve in red, and a line with a slope of
132.5335679 meeting the elliptic curve at P2, P2 has an x value of -0.0248588.
I don't know what you mean by "value of the cusp". The elliptic curve and the
black line meet at an angle of zero, which would fit the aeronautical
engineering definition of "cusp".

The lower graph shows a meeting of the elliptic curve and the black line without
a cusp.

Are you able to see the graphs? Do you not agree that the upper graph shows a
cusp, and the lower graph does not? The definition from my Dictionary of
Mathematics applies to the top graph, but not the bottom graph.


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".

See just above.


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.

In that case, the "analytical" curve can leave the elliptic curve at P2 in two
ways. It can leave going to the northeast, or it can leave going to the
southwest. Either can satisfy the slope requirement, but one case produces a
cusp (the top graph at the URL given above) and the other does not (bottom
graph).


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

Better is in the eye of the beholder. The cubic curve discussed earlier meets
P2 travelling northeasterly. Your elliptic curve meets P2 travelling
southwesterly. Does the difference not matter to you? If not, then the
elliptic curve may be "better" for your purposes.


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

Monir

.

Relevant Pages

  • Re: Unstable Simultaneous Linear Equations!
    ... "Do you not agree that the upper graph shows a cusp, and the lower graph does not? ... I decided earlier on to display the "slope line" at P2 just to show ... that the elliptic curve satisfies the slope condition and smoothly ...
    (sci.math.num-analysis)
  • Re: Unstable Simultaneous Linear Equations!
    ... but then you say that you want to find a curve that has the proper derivative AT ... It would be EXTREMELY helpful if you would show, in a graph, the analytical ... and minimum were part of your elliptic curve. ... "Cusp, A double point at which the two tangents to the curve are coincident. ...
    (sci.math.num-analysis)
  • Re: Unstable Simultaneous Linear Equations!
    ... These points are on the very right end of the ellipse and there is no cusp ... associated with this curve BY ITSELF. ... I have assumed that you want to be going northeast when you reach P2. ... elliptic curve, you must reverse direction when you reach P2 in order to ...
    (sci.math.num-analysis)
  • Re: Unstable Simultaneous Linear Equations!
    ... independent variable "x" in relation to the centre of the curve and ... ellipse axis along the X-axis = 2a ... and you want>to fill the gap with an interpolating curve. ... This creates a cusp. ...
    (sci.math.num-analysis)
  • Re: Unstable Simultaneous Linear Equations!
    ... independent variable "x" in relation to the centre of the curve and ... an analytic development between two points, ... ellipse axis along the X-axis = 2a ... This creates a cusp. ...
    (sci.math.num-analysis)