There are 4 coefficients in a 3rd-order equation. If you have more than 4 data points, they usually do not fit. A very common situation is, the data points have measurement errors. And you want to find the best fit to minimize the deviations from these measured data points.

The algorithm to do this was developed way before we have computers and calculators. It was used very commonly when computers and calculator were used for computing.

Look for Least Square Method.

--- In m...@yahoogroups.com, "Mike" wrote:
>
> Hi,
>
> I'm looking for C-code to generate a 3rd-order equation from a set of (typically) 5 - 9 data points. For example, given x,y points:
>
> {-.04, -.0007}
> {-.03, -.0003}
> {02, -0.00008}
> {-.01, -0.00001}
> {0, 0}
> {.01, 0.00001}
> {.02, 0.00007}
> {.03, .0002}
> {.04, .0006}
>
> I should end up with an equation like this (so Excel tells me):
>
> y = 10.581x^3 - 0.037x^2 - 0.0008x + 1E-06
>
> After researching the math on the internet, my code for a 3rd order equation doesn't get anywhere close to this. Not sure if it's a bug making the linear equations from the points, or solving the equations. I trust my coding skills, but not my math... I'm using IAR's floating point lbrary (sorry Al!).
>
> Does anyone have a routine handy that would do this? Or know of a math package that I can buy that will let me extract the C-code for just this without having to include the whole library?
>
> Thanks,
> Mike.
> Kaelin Consulting
> mike@...
>

------------------------------------



(You need to be a member of msp430 -- send a blank email to msp430-subscribe@yahoogroups.com )

On Tue, 14 Jul 2009 13:03:02 -0000, you wrote:

>I'm looking for C-code to generate a 3rd-order equation from a set of
>(typically) 5 - 9 data points. For example, given x,y points:
>
>{-.04, -.0007}
>{-.03, -.0003}
>{02, -0.00008}
>{-.01, -0.00001}
>{0, 0}
>{.01, 0.00001}
>{.02, 0.00007}
>{.03, .0002}
>{.04, .0006}
>
>I should end up with an equation like this (so Excel tells me):
>
>y = 10.581x^3 - 0.037x^2 - 0.0008x + 1E-06
>
>After researching the math on the internet, my code for a 3rd order
>equation doesn't get anywhere close to this. Not sure if it's a bug
>making the linear equations from the points, or solving the
>equations. I trust my coding skills, but not my math... I'm using
>IAR's floating point lbrary (sorry Al!).
>
>Does anyone have a routine handy that would do this? Or know of a math
>package that I can buy that will let me extract the C-code for just
>this without having to include the whole library?

I've not done this, but it would simply be a matter of setting up the
error term (probably just error^2 so that it is positive-valued, but
there are other possibilities) and the partial differential equations
and solving the simultaneous set. I haven't done it before.

The error term would be:

SUM_OVER_DATA of (Y_i - a0 - a1*X_i - a2*X_i^2 - a3*X_i^3)^2

if that makes sense to you. It's just the difference between the Y
value and the computed equation's value using X, instead. That get's
squared beforing summing them up. You will want to minimize that
error term, which is why you want to take partial derivatives of it
with respect to each of the constants, a0, a1, a2, and a3, and set
those to zero. Solving that set will give you what you need to know.

Some good numerical methods books will walk you through the process,
generally. Numerical Recipes also does this for the generalized
version of the above equation. They even include the possibility of
using known measurement errors in the computation (which weights
differently, different values, if you like.) If you don't have the
book, get it. A worked routine will be in there.

Also, I see this (have not looked it over):

http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearleastsquaresregression_8cpp-source.html
http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearleastsquaresregression_8hpp-source.html

This is an excellent opportunity for you to learn about this useful
and powerful technique so that it is something you own for yourself
and no one can ever take it away from you, again.

Jon
------------------------------------



(You need to be a member of msp430 -- send a blank email to msp430-subscribe@yahoogroups.com )

Google "least squares fit polynomial":

http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html

The algorithm is very easy to solve if your C library supplies basic
linear algebra. Otherwise it's a bunch of for loops.

On Tue, Jul 14, 2009 at 9:02 AM, Jon Kirwan wrote:
> On Tue, 14 Jul 2009 13:03:02 -0000, you wrote:
>
>>I'm looking for C-code to generate a 3rd-order equation from a set of
>>(typically) 5 - 9 data points. For example, given x,y points:
>>
>>{-.04, -.0007}
>>{-.03, -.0003}
>>{02, -0.00008}
>>{-.01, -0.00001}
>>{0, 0}
>>{.01, 0.00001}
>>{.02, 0.00007}
>>{.03, .0002}
>>{.04, .0006}
>>
>>I should end up with an equation like this (so Excel tells me):
>>
>>y = 10.581x^3 - 0.037x^2 - 0.0008x + 1E-06
>>
>>After researching the math on the internet, my code for a 3rd order
>>equation doesn't get anywhere close to this. Not sure if it's a bug
>>making the linear equations from the points, or solving the
>>equations. I trust my coding skills, but not my math... I'm using
>>IAR's floating point lbrary (sorry Al!).
>>
>>Does anyone have a routine handy that would do this? Or know of a math
>>package that I can buy that will let me extract the C-code for just
>>this without having to include the whole library?
>
> I've not done this, but it would simply be a matter of setting up the
> error term (probably just error^2 so that it is positive-valued, but
> there are other possibilities) and the partial differential equations
> and solving the simultaneous set. I haven't done it before.
>
> The error term would be:
>
> SUM_OVER_DATA of (Y_i - a0 - a1*X_i - a2*X_i^2 - a3*X_i^3)^2
>
> if that makes sense to you. It's just the difference between the Y
> value and the computed equation's value using X, instead. That get's
> squared beforing summing them up. You will want to minimize that
> error term, which is why you want to take partial derivatives of it
> with respect to each of the constants, a0, a1, a2, and a3, and set
> those to zero. Solving that set will give you what you need to know.
>
> Some good numerical methods books will walk you through the process,
> generally. Numerical Recipes also does this for the generalized
> version of the above equation. They even include the possibility of
> using known measurement errors in the computation (which weights
> differently, different values, if you like.) If you don't have the
> book, get it. A worked routine will be in there.
>
> Also, I see this (have not looked it over):
>
> http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearleastsquaresregression_8cpp-source.html
> http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearleastsquaresregression_8hpp-source.html
>
> This is an excellent opportunity for you to learn about this useful
> and powerful technique so that it is something you own for yourself
> and no one can ever take it away from you, again.
>
> Jon
------------------------------------



(You need to be a member of msp430 -- send a blank email to msp430-subscribe@yahoogroups.com )

If your math isn't so good to interpret the link I sent, here more of a hin=
t:

given n points with n>3
(x1,y1), (x2,y2), ... (xn,yn)

then find the the coefficients a0, a1, a2 such that

y=3Da0+a1*x+a2*x^2 is the equation you want

so eqn (15) of the link says:

[a0; a1; a2]=3D inv(Xt*X) * Xt * [y1; y2; ... yn]

where
X=3D[1 x1 x1^2; 1 x2 x2^2; ... 1 xn xn^2]

so just compute the transpose, matrix multiplications and the inv operation=
s.

Again very easy if you have some basic linear algebra library or time
to code it.

Henry

On Wed, Jul 15, 2009 at 6:08 PM, Henry Liu wrote:
> Google "least squares fit polynomial":
>
> http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html
>
> The algorithm is very easy to solve if your C library supplies basic
> linear algebra. =A0Otherwise it's a bunch of for loops.
>
> On Tue, Jul 14, 2009 at 9:02 AM, Jon Kirwan wro=
te:
>> On Tue, 14 Jul 2009 13:03:02 -0000, you wrote:
>>
>>>I'm looking for C-code to generate a 3rd-order equation from a set of
>>>(typically) 5 - 9 data points. For example, given x,y points:
>>>
>>>{-.04, -.0007}
>>>{-.03, -.0003}
>>>{02, -0.00008}
>>>{-.01, -0.00001}
>>>{0, 0}
>>>{.01, 0.00001}
>>>{.02, 0.00007}
>>>{.03, .0002}
>>>{.04, .0006}
>>>
>>>I should end up with an equation like this (so Excel tells me):
>>>
>>>y =3D 10.581x^3 - 0.037x^2 - 0.0008x + 1E-06
>>>
>>>After researching the math on the internet, my code for a 3rd order
>>>equation doesn't get anywhere close to this. Not sure if it's a bug
>>>making the linear equations from the points, or solving the
>>>equations. I trust my coding skills, but not my math... I'm using
>>>IAR's floating point lbrary (sorry Al!).
>>>
>>>Does anyone have a routine handy that would do this? Or know of a math
>>>package that I can buy that will let me extract the C-code for just
>>>this without having to include the whole library?
>>
>> I've not done this, but it would simply be a matter of setting up the
>> error term (probably just error^2 so that it is positive-valued, but
>> there are other possibilities) and the partial differential equations
>> and solving the simultaneous set. I haven't done it before.
>>
>> The error term would be:
>>
>> SUM_OVER_DATA of (Y_i - a0 - a1*X_i - a2*X_i^2 - a3*X_i^3)^2
>>
>> if that makes sense to you. It's just the difference between the Y
>> value and the computed equation's value using X, instead. That get's
>> squared beforing summing them up. You will want to minimize that
>> error term, which is why you want to take partial derivatives of it
>> with respect to each of the constants, a0, a1, a2, and a3, and set
>> those to zero. Solving that set will give you what you need to know.
>>
>> Some good numerical methods books will walk you through the process,
>> generally. Numerical Recipes also does this for the generalized
>> version of the above equation. They even include the possibility of
>> using known measurement errors in the computation (which weights
>> differently, different values, if you like.) If you don't have the
>> book, get it. A worked routine will be in there.
>>
>> Also, I see this (have not looked it over):
>>
>> http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearl=
eastsquaresregression_8cpp-source.html
>> http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearl=
eastsquaresregression_8hpp-source.html
>>
>> This is an excellent opportunity for you to learn about this useful
>> and powerful technique so that it is something you own for yourself
>> and no one can ever take it away from you, again.
>>
>> Jon
>>
>>=20
>
------------------------------------



(You need to be a member of msp430 -- send a blank email to msp430-subscribe@yahoogroups.com )

On Wed, 15 Jul 2009 18:08:01 -0700, you wrote:

>Google "least squares fit polynomial":
>
>http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html
>
>The algorithm is very easy to solve if your C library supplies basic
>linear algebra. Otherwise it's a bunch of for loops.

Henry, if you are helping the OP, fine. But it may be a little
confusing to reply to me, if that is the case. It sounds like you are
trying to help the wrong person the way you wrote your reply. I don't
need it as I can easily do the partials by myself and solve the
resulting set. (And I've coded my own command line linear solver
whenever the need arises for that.)

Jon
------------------------------------



(You need to be a member of msp430 -- send a blank email to msp430-subscribe@yahoogroups.com )

> y=3Da0+a1*x+a2*x^2 is the equation you want

I don't get this...
An equation like that :
ax^2 + bx + c
produces a parabolic plot.
So how can that fit into only a few data points ?
Or is the inverse method, using just a few data points as input just a roug=
h approximation
?

Best Regards,
Kris=20

> -----Original Message-----
> From: m...@yahoogroups.com [mailto:m...@yahoogroups.com] On Behalf Of=
Henry Liu
> Sent: Thursday, 16 July 2009 11:18 AM
> To: m...@yahoogroups.com
> Subject: Re: [msp430] creating and solving 3rd order equations from data =
points
>=20
> If your math isn't so good to interpret the link I sent, here more of a h=
int:
>=20
> given n points with n>3
> (x1,y1), (x2,y2), ... (xn,yn)
>=20
> then find the the coefficients a0, a1, a2 such that
>=20
> y=3Da0+a1*x+a2*x^2 is the equation you want
>=20
> so eqn (15) of the link says:
>=20
> [a0; a1; a2]=3D inv(Xt*X) * Xt * [y1; y2; ... yn]
>=20
> where
> X=3D[1 x1 x1^2; 1 x2 x2^2; ... 1 xn xn^2]
>=20
> so just compute the transpose, matrix multiplications and the inv operati=
ons.
>=20
> Again very easy if you have some basic linear algebra library or time
> to code it.
>=20
> Henry
>=20
> On Wed, Jul 15, 2009 at 6:08 PM, Henry Liu wrote:
> > Google "least squares fit polynomial":
> >
> > http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html
> >
> > The algorithm is very easy to solve if your C library supplies basic
> > linear algebra. =A0Otherwise it's a bunch of for loops.
> >
> > On Tue, Jul 14, 2009 at 9:02 AM, Jon Kirwan w=
rote:
> >>
> >>
> >> On Tue, 14 Jul 2009 13:03:02 -0000, you wrote:
> >>
> >>>I'm looking for C-code to generate a 3rd-order equation from a set of
> >>>(typically) 5 - 9 data points. For example, given x,y points:
> >>>
> >>>{-.04, -.0007}
> >>>{-.03, -.0003}
> >>>{02, -0.00008}
> >>>{-.01, -0.00001}
> >>>{0, 0}
> >>>{.01, 0.00001}
> >>>{.02, 0.00007}
> >>>{.03, .0002}
> >>>{.04, .0006}
> >>>
> >>>I should end up with an equation like this (so Excel tells me):
> >>>
> >>>y =3D 10.581x^3 - 0.037x^2 - 0.0008x + 1E-06
> >>>
> >>>After researching the math on the internet, my code for a 3rd order
> >>>equation doesn't get anywhere close to this. Not sure if it's a bug
> >>>making the linear equations from the points, or solving the
> >>>equations. I trust my coding skills, but not my math... I'm using
> >>>IAR's floating point lbrary (sorry Al!).
> >>>
> >>>Does anyone have a routine handy that would do this? Or know of a math
> >>>package that I can buy that will let me extract the C-code for just
> >>>this without having to include the whole library?
> >>
> >> I've not done this, but it would simply be a matter of setting up the
> >> error term (probably just error^2 so that it is positive-valued, but
> >> there are other possibilities) and the partial differential equations
> >> and solving the simultaneous set. I haven't done it before.
> >>
> >> The error term would be:
> >>
> >> SUM_OVER_DATA of (Y_i - a0 - a1*X_i - a2*X_i^2 - a3*X_i^3)^2
> >>
> >> if that makes sense to you. It's just the difference between the Y
> >> value and the computed equation's value using X, instead. That get's
> >> squared beforing summing them up. You will want to minimize that
> >> error term, which is why you want to take partial derivatives of it
> >> with respect to each of the constants, a0, a1, a2, and a3, and set
> >> those to zero. Solving that set will give you what you need to know.
> >>
> >> Some good numerical methods books will walk you through the process,
> >> generally. Numerical Recipes also does this for the generalized
> >> version of the above equation. They even include the possibility of
> >> using known measurement errors in the computation (which weights
> >> differently, different values, if you like.) If you don't have the
> >> book, get it. A worked routine will be in there.
> >>
> >> Also, I see this (have not looked it over):
> >>
> >http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearleas=
tsquaresregress
ion_8
> cpp-source.html
> >http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearleas=
tsquaresregress
ion_8
> hpp-source.html
> >>
> >> This is an excellent opportunity for you to learn about this useful
> >> and powerful technique so that it is something you own for yourself
> >> and no one can ever take it away from you, again.
> >>
> >> Jon
> >>
> >>
> >
>=20
>=20
> ------------------------------------
>=20
>
>=20
>



(You need to be a member of msp430 -- send a blank email to msp430-subscribe@yahoogroups.com )

Least Square Method is commonly used in Regression Analysis.=20
Perhaps we regressed ;-)

--- In m...@yahoogroups.com, "Microbit_P43000" wrote:
>
> > y=3Da0+a1*x+a2*x^2 is the equation you want
>=20
> I don't get this...
> An equation like that :
> ax^2 + bx + c
> produces a parabolic plot.
> So how can that fit into only a few data points ?
> Or is the inverse method, using just a few data points as input just a ro=
ugh approximation
> ?
>=20
> Best Regards,
> Kris=20
>=20
> > -----Original Message-----
> > From: m...@yahoogroups.com [mailto:m...@yahoogroups.com] On Behalf =
Of Henry Liu
> > Sent: Thursday, 16 July 2009 11:18 AM
> > To: m...@yahoogroups.com
> > Subject: Re: [msp430] creating and solving 3rd order equations from dat=
a points
> >=20
> > If your math isn't so good to interpret the link I sent, here more of a=
hint:
> >=20
> > given n points with n>3
> > (x1,y1), (x2,y2), ... (xn,yn)
> >=20
> > then find the the coefficients a0, a1, a2 such that
> >=20
> > y=3Da0+a1*x+a2*x^2 is the equation you want
> >=20
> > so eqn (15) of the link says:
> >=20
> > [a0; a1; a2]=3D inv(Xt*X) * Xt * [y1; y2; ... yn]
> >=20
> > where
> > X=3D[1 x1 x1^2; 1 x2 x2^2; ... 1 xn xn^2]
> >=20
> > so just compute the transpose, matrix multiplications and the inv opera=
tions.
> >=20
> > Again very easy if you have some basic linear algebra library or time
> > to code it.
> >=20
> > Henry
> >=20
> > On Wed, Jul 15, 2009 at 6:08 PM, Henry Liu wrote:
> > > Google "least squares fit polynomial":
> > >
> > > http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html
> > >
> > > The algorithm is very easy to solve if your C library supplies basic
> > > linear algebra. =A0Otherwise it's a bunch of for loops.
> > >
> > > On Tue, Jul 14, 2009 at 9:02 AM, Jon Kirwan wrote:
> > >>
> > >>
> > >> On Tue, 14 Jul 2009 13:03:02 -0000, you wrote:
> > >>
> > >>>I'm looking for C-code to generate a 3rd-order equation from a set o=
f
> > >>>(typically) 5 - 9 data points. For example, given x,y points:
> > >>>
> > >>>{-.04, -.0007}
> > >>>{-.03, -.0003}
> > >>>{02, -0.00008}
> > >>>{-.01, -0.00001}
> > >>>{0, 0}
> > >>>{.01, 0.00001}
> > >>>{.02, 0.00007}
> > >>>{.03, .0002}
> > >>>{.04, .0006}
> > >>>
> > >>>I should end up with an equation like this (so Excel tells me):
> > >>>
> > >>>y =3D 10.581x^3 - 0.037x^2 - 0.0008x + 1E-06
> > >>>
> > >>>After researching the math on the internet, my code for a 3rd order
> > >>>equation doesn't get anywhere close to this. Not sure if it's a bug
> > >>>making the linear equations from the points, or solving the
> > >>>equations. I trust my coding skills, but not my math... I'm using
> > >>>IAR's floating point lbrary (sorry Al!).
> > >>>
> > >>>Does anyone have a routine handy that would do this? Or know of a ma=
th
> > >>>package that I can buy that will let me extract the C-code for just
> > >>>this without having to include the whole library?
> > >>
> > >> I've not done this, but it would simply be a matter of setting up th=
e
> > >> error term (probably just error^2 so that it is positive-valued, but
> > >> there are other possibilities) and the partial differential equation=
s
> > >> and solving the simultaneous set. I haven't done it before.
> > >>
> > >> The error term would be:
> > >>
> > >> SUM_OVER_DATA of (Y_i - a0 - a1*X_i - a2*X_i^2 - a3*X_i^3)^2
> > >>
> > >> if that makes sense to you. It's just the difference between the Y
> > >> value and the computed equation's value using X, instead. That get's
> > >> squared beforing summing them up. You will want to minimize that
> > >> error term, which is why you want to take partial derivatives of it
> > >> with respect to each of the constants, a0, a1, a2, and a3, and set
> > >> those to zero. Solving that set will give you what you need to know.
> > >>
> > >> Some good numerical methods books will walk you through the process,
> > >> generally. Numerical Recipes also does this for the generalized
> > >> version of the above equation. They even include the possibility of
> > >> using known measurement errors in the computation (which weights
> > >> differently, different values, if you like.) If you don't have the
> > >> book, get it. A worked routine will be in there.
> > >>
> > >> Also, I see this (have not looked it over):
> > >>
> > >>
> >
> http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearle=
astsquaresregress
> ion_8
> > cpp-source.html
> > >>
> >
> http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearle=
astsquaresregress
> ion_8
> > hpp-source.html
> > >>
> > >> This is an excellent opportunity for you to learn about this useful
> > >> and powerful technique so that it is something you own for yourself
> > >> and no one can ever take it away from you, again.
> > >>
> > >> Jon
> > >>
> > >>
> > >
> >=20
> >=20
> > ------------------------------------
> >=20
> >
> >=20
> >



(You need to be a member of msp430 -- send a blank email to msp430-subscribe@yahoogroups.com )

Hi Henry,=20

I saw the website you listed, plus several others... Let me walk through th=
is to verify my thinking...=20

For linear regression, as I understand it, the matrix looks like this:

[
n sum(x) sum(x^2) sum(x^3) -sum(y)

sum(x) sum(x^2) sum(x^3) sum(x^4) -sum(x)sum(y)

sum(x^2) sum(x^3) sum(x^4) sum(x^5) -sum(x^2)sum(y)

sum(x^3) sum(x^4) sum(x^5) sum(x^6) -sum(x^3)sum(y)
]

Where n is the number of points, sum(x) is the sum of all x coordinates, su=
m(x^2) is the sum of all x coordinates squared, etc.=20

Using Gaussian elimination (again, as I understand it), the matrix I end up=
after reduction looks like this:

1 0 0 0 A
0 1 0 0 B
0 0 1 0 C
0 0 0 1 D

Meaning that y =3D Ax + Bx^2 + Cx^3 + D

As I said in the previous post, A,B,C,D don't look anything like what I get=
out of Excel. Unfortunately, Excel doesn't give me intermediate results so=
that I can verify my equations. If you see a problem with my math or logic=
, I can start there... Otherwise I can look at the mechanics of my C-code f=
or an error. I appreciate your help!

Thanks,
Mike.

--- In m...@yahoogroups.com, Henry Liu wrote:
>
> If your math isn't so good to interpret the link I sent, here more of a h=
int:
>=20
> given n points with n>3
> (x1,y1), (x2,y2), ... (xn,yn)
>=20
> then find the the coefficients a0, a1, a2 such that
>=20
> y=3Da0+a1*x+a2*x^2 is the equation you want
>=20
> so eqn (15) of the link says:
>=20
> [a0; a1; a2]=3D inv(Xt*X) * Xt * [y1; y2; ... yn]
>=20
> where
> X=3D[1 x1 x1^2; 1 x2 x2^2; ... 1 xn xn^2]
>=20
> so just compute the transpose, matrix multiplications and the inv operati=
ons.
>=20
> Again very easy if you have some basic linear algebra library or time
> to code it.
>=20
> Henry
>=20
> On Wed, Jul 15, 2009 at 6:08 PM, Henry Liu wrote:
> > Google "least squares fit polynomial":
> >
> > http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html
> >
> > The algorithm is very easy to solve if your C library supplies basic
> > linear algebra. =A0Otherwise it's a bunch of for loops.
> >
> > On Tue, Jul 14, 2009 at 9:02 AM, Jon Kirwan wrote:
> >>
> >>
> >> On Tue, 14 Jul 2009 13:03:02 -0000, you wrote:
> >>
> >>>I'm looking for C-code to generate a 3rd-order equation from a set of
> >>>(typically) 5 - 9 data points. For example, given x,y points:
> >>>
> >>>{-.04, -.0007}
> >>>{-.03, -.0003}
> >>>{02, -0.00008}
> >>>{-.01, -0.00001}
> >>>{0, 0}
> >>>{.01, 0.00001}
> >>>{.02, 0.00007}
> >>>{.03, .0002}
> >>>{.04, .0006}
> >>>
> >>>I should end up with an equation like this (so Excel tells me):
> >>>
> >>>y =3D 10.581x^3 - 0.037x^2 - 0.0008x + 1E-06
> >>>
> >>>After researching the math on the internet, my code for a 3rd order
> >>>equation doesn't get anywhere close to this. Not sure if it's a bug
> >>>making the linear equations from the points, or solving the
> >>>equations. I trust my coding skills, but not my math... I'm using
> >>>IAR's floating point lbrary (sorry Al!).
> >>>
> >>>Does anyone have a routine handy that would do this? Or know of a math
> >>>package that I can buy that will let me extract the C-code for just
> >>>this without having to include the whole library?
> >>

------------------------------------



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Sorry for third order it's ax^3+bx^2+cx+d

Easy enough change

On Wed, Jul 15, 2009 at 9:48 PM,
Microbit_P43000 wrote:
>> y=3Da0+a1*x+a2*x^2 is the equation you want
>
> I don't get this...
> An equation like that :
> ax^2 + bx + c
> produces a parabolic plot.
> So how can that fit into only a few data points ?
> Or is the inverse method, using just a few data points as input just a ro=
ugh
> approximation
> ?
>
> Best Regards,
> Kris
>
>> -----Original Message-----
>> From: m...@yahoogroups.com [mailto:m...@yahoogroups.com] On Behalf O=
f
>> Henry Liu
>> Sent: Thursday, 16 July 2009 11:18 AM
>> To: m...@yahoogroups.com
>> Subject: Re: [msp430] creating and solving 3rd order equations from data
>> points
>>
>> If your math isn't so good to interpret the link I sent, here more of a
>> hint:
>>
>> given n points with n>3
>> (x1,y1), (x2,y2), ... (xn,yn)
>>
>> then find the the coefficients a0, a1, a2 such that
>>
>> y=3Da0+a1*x+a2*x^2 is the equation you want
>>
>> so eqn (15) of the link says:
>>
>> [a0; a1; a2]=3D inv(Xt*X) * Xt * [y1; y2; ... yn]
>>
>> where
>> X=3D[1 x1 x1^2; 1 x2 x2^2; ... 1 xn xn^2]
>>
>> so just compute the transpose, matrix multiplications and the inv
>> operations.
>>
>> Again very easy if you have some basic linear algebra library or time
>> to code it.
>>
>> Henry
>>
>> On Wed, Jul 15, 2009 at 6:08 PM, Henry Liu wrote:
>> > Google "least squares fit polynomial":
>> >
>> > http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html
>> >
>> > The algorithm is very easy to solve if your C library supplies basic
>> > linear algebra. =A0Otherwise it's a bunch of for loops.
>> >
>> > On Tue, Jul 14, 2009 at 9:02 AM, Jon Kirwan
>> > wrote:
>> >>
>> >>
>> >> On Tue, 14 Jul 2009 13:03:02 -0000, you wrote:
>> >>
>> >>>I'm looking for C-code to generate a 3rd-order equation from a set of
>> >>>(typically) 5 - 9 data points. For example, given x,y points:
>> >>>
>> >>>{-.04, -.0007}
>> >>>{-.03, -.0003}
>> >>>{02, -0.00008}
>> >>>{-.01, -0.00001}
>> >>>{0, 0}
>> >>>{.01, 0.00001}
>> >>>{.02, 0.00007}
>> >>>{.03, .0002}
>> >>>{.04, .0006}
>> >>>
>> >>>I should end up with an equation like this (so Excel tells me):
>> >>>
>> >>>y =3D 10.581x^3 - 0.037x^2 - 0.0008x + 1E-06
>> >>>
>> >>>After researching the math on the internet, my code for a 3rd order
>> >>>equation doesn't get anywhere close to this. Not sure if it's a bug
>> >>>making the linear equations from the points, or solving the
>> >>>equations. I trust my coding skills, but not my math... I'm using
>> >>>IAR's floating point lbrary (sorry Al!).
>> >>>
>> >>>Does anyone have a routine handy that would do this? Or know of a mat=
h
>> >>>package that I can buy that will let me extract the C-code for just
>> >>>this without having to include the whole library?
>> >>
>> >> I've not done this, but it would simply be a matter of setting up the
>> >> error term (probably just error^2 so that it is positive-valued, but
>> >> there are other possibilities) and the partial differential equations
>> >> and solving the simultaneous set. I haven't done it before.
>> >>
>> >> The error term would be:
>> >>
>> >> SUM_OVER_DATA of (Y_i - a0 - a1*X_i - a2*X_i^2 - a3*X_i^3)^2
>> >>
>> >> if that makes sense to you. It's just the difference between the Y
>> >> value and the computed equation's value using X, instead. That get's
>> >> squared beforing summing them up. You will want to minimize that
>> >> error term, which is why you want to take partial derivatives of it
>> >> with respect to each of the constants, a0, a1, a2, and a3, and set
>> >> those to zero. Solving that set will give you what you need to know.
>> >>
>> >> Some good numerical methods books will walk you through the process,
>> >> generally. Numerical Recipes also does this for the generalized
>> >> version of the above equation. They even include the possibility of
>> >> using known measurement errors in the computation (which weights
>> >> differently, different values, if you like.) If you don't have the
>> >> book, get it. A worked routine will be in there.
>> >>
>> >> Also, I see this (have not looked it over):
>> >>
>> > http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearle=
astsquaresregress
> ion_8
>> cpp-source.html
>> > http://quantlib.sourcearchive.com/documentation/0.3.13/ql_2Math_2linearle=
astsquaresregress
> ion_8
>> hpp-source.html
>> >>
>> >> This is an excellent opportunity for you to learn about this useful
>> >> and powerful technique so that it is something you own for yourself
>> >> and no one can ever take it away from you, again.
>> >>
>> >> Jon
>> >>
>> >>
>> >
>> ------------------------------------



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I think you flipped a minus sign:

http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html

eqn 9 is equiv to running Guass jordan elimination on:

[
n sum(x) sum(x^2) sum(x^3) sum(y)

sum(x) sum(x^2) sum(x^3) sum(x^4) sum(x)sum(y)

sum(x^2) sum(x^3) sum(x^4) sum(x^5) sum(x^2)sum(y)

sum(x^3) sum(x^4) sum(x^5) sum(x^6) sum(x^3)sum(y)
]

On Thu, Jul 16, 2009 at 8:46 AM, Mike wrote:
> Hi Henry,
>
> I saw the website you listed, plus several others... Let me walk through
> this to verify my thinking...
>
> For linear regression, as I understand it, the matrix looks like this:
>
> [
> n sum(x) sum(x^2) sum(x^3) -sum(y)
>
> sum(x) sum(x^2) sum(x^3) sum(x^4) -sum(x)sum(y)
>
> sum(x^2) sum(x^3) sum(x^4) sum(x^5) -sum(x^2)sum(y)
>
> sum(x^3) sum(x^4) sum(x^5) sum(x^6) -sum(x^3)sum(y)
> ]
>
> Where n is the number of points, sum(x) is the sum of all x coordinates,
> sum(x^2) is the sum of all x coordinates squared, etc.
>
> Using Gaussian elimination (again, as I understand it), the matrix I end =
up
> after reduction looks like this:
>
> 1 0 0 0 A
> 0 1 0 0 B
> 0 0 1 0 C
> 0 0 0 1 D
>
> Meaning that y =3D Ax + Bx^2 + Cx^3 + D
>
> As I said in the previous post, A,B,C,D don't look anything like what I g=
et
> out of Excel. Unfortunately, Excel doesn't give me intermediate results s=
o
> that I can verify my equations. If you see a problem with my math or logi=
c,
> I can start there... Otherwise I can look at the mechanics of my C-code f=
or
> an error. I appreciate your help!
>
> Thanks,
> Mike.
>
> --- In m...@yahoogroups.com, Henry Liu wrote:
>>
>> If your math isn't so good to interpret the link I sent, here more of a
>> hint:
>>
>> given n points with n>3
>> (x1,y1), (x2,y2), ... (xn,yn)
>>
>> then find the the coefficients a0, a1, a2 such that
>>
>> y=3Da0+a1*x+a2*x^2 is the equation you want
>>
>> so eqn (15) of the link says:
>>
>> [a0; a1; a2]=3D inv(Xt*X) * Xt * [y1; y2; ... yn]
>>
>> where
>> X=3D[1 x1 x1^2; 1 x2 x2^2; ... 1 xn xn^2]
>>
>> so just compute the transpose, matrix multiplications and the inv
>> operations.
>>
>> Again very easy if you have some basic linear algebra library or time
>> to code it.
>>
>> Henry
>>
>> On Wed, Jul 15, 2009 at 6:08 PM, Henry Liu wrote:
>> > Google "least squares fit polynomial":
>> >
>> > http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html
>> >
>> > The algorithm is very easy to solve if your C library supplies basic
>> > linear algebra. =A0Otherwise it's a bunch of for loops.
>> >
>> > On Tue, Jul 14, 2009 at 9:02 AM, Jon Kirwan wrote:
>> >>
>> >>
>> >> On Tue, 14 Jul 2009 13:03:02 -0000, you wrote:
>> >>
>> >>>I'm looking for C-code to generate a 3rd-order equation from a set of
>> >>>(typically) 5 - 9 data points. For example, given x,y points:
>> >>>
>> >>>{-.04, -.0007}
>> >>>{-.03, -.0003}
>> >>>{02, -0.00008}
>> >>>{-.01, -0.00001}
>> >>>{0, 0}
>> >>>{.01, 0.00001}
>> >>>{.02, 0.00007}
>> >>>{.03, .0002}
>> >>>{.04, .0006}
>> >>>
>> >>>I should end up with an equation like this (so Excel tells me):
>> >>>
>> >>>y =3D 10.581x^3 - 0.037x^2 - 0.0008x + 1E-06
>> >>>
>> >>>After researching the math on the internet, my code for a 3rd order
>> >>>equation doesn't get anywhere close to this. Not sure if it's a bug
>> >>>making the linear equations from the points, or solving the
>> >>>equations. I trust my coding skills, but not my math... I'm using
>> >>>IAR's floating point lbrary (sorry Al!).
>> >>>
>> >>>Does anyone have a routine handy that would do this? Or know of a mat=
h
>> >>>package that I can buy that will let me extract the C-code for just
>> >>>this without having to include the whole library?
>> >>=20
------------------------------------

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