Given four points how can I find an equation for any pattern?Explanation of Lagrange Interpolating...
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Given four points how can I find an equation for any pattern?
Explanation of Lagrange Interpolating PolynomialHow to find what a given series/sequence converges toA complex sequence that has $n$ limit points, for any natural number $n$;How do you find such points?how can I find roots of $x^2+px+q=0$ using iterative methods?How to find the fixed points of $sin(1/x)$?Any function that equals $0$ a.e. implies Lebesgue integral also equals $0$How can we find the largest $B$ that the implications of the implicit function theorem hold?How to find radius of convergence of given rational function?How to find lower Riemann integral in given function?How can I prove that for any n≥1, n points can be found in $mathcal{C}[0,1]$ such that in the d metric, the distance between any two points equals 1?
$begingroup$
If for example I know that there's a function f(x) that equals 1 at x=8, 8 at x=32, 36 at x=64, 98 at x=128.
How can I find the expression for this pattern or any other pattern?
real-analysis
asked 2 hours ago
Mohammad AlSaqqaMohammad AlSaqqa
61
New contributor
Mohammad AlSaqqa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
If for example I know that there's a function f(x) that equals 1 at x=8, 8 at x=32, 36 at x=64, 98 at x=128.
How can I find the expression for this pattern or any other pattern?
real-analysis
asked 2 hours ago
Mohammad AlSaqqaMohammad AlSaqqa
61
New contributor
Mohammad AlSaqqa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
2
$begingroup$
You're not guaranteed one unique pattern/function. You can use polynomial interpolation to get a formula $f$ where $deg(f) = n-1$ if you have have $n$ pairs $(x,f(x))$. Said $f$ will even be unique. But higher degree polynomials could also fit it, infinitely many in fact. And this is all just polynomial functions mind you.
$endgroup$
– Eevee Trainer
2 hours ago
add a comment |
$begingroup$
If for example I know that there's a function f(x) that equals 1 at x=8, 8 at x=32, 36 at x=64, 98 at x=128.
How can I find the expression for this pattern or any other pattern?
real-analysis
asked 2 hours ago
Mohammad AlSaqqaMohammad AlSaqqa
61
New contributor
Mohammad AlSaqqa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
If for example I know that there's a function f(x) that equals 1 at x=8, 8 at x=32, 36 at x=64, 98 at x=128.
How can I find the expression for this pattern or any other pattern?
real-analysis
real-analysis
asked 2 hours ago
Mohammad AlSaqqaMohammad AlSaqqa
61
New contributor
Mohammad AlSaqqa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
Mohammad AlSaqqaMohammad AlSaqqa
61
New contributor
Mohammad AlSaqqa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
Mohammad AlSaqqaMohammad AlSaqqa
61
New contributor
Mohammad AlSaqqa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
Mohammad AlSaqqaMohammad AlSaqqa
61
asked 2 hours ago
Mohammad AlSaqqaMohammad AlSaqqa
61
61
New contributor
Mohammad AlSaqqa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Mohammad AlSaqqa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Mohammad AlSaqqa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
$begingroup$
You're not guaranteed one unique pattern/function. You can use polynomial interpolation to get a formula $f$ where $deg(f) = n-1$ if you have have $n$ pairs $(x,f(x))$. Said $f$ will even be unique. But higher degree polynomials could also fit it, infinitely many in fact. And this is all just polynomial functions mind you.
$endgroup$
– Eevee Trainer
2 hours ago
add a comment |
2
$begingroup$
You're not guaranteed one unique pattern/function. You can use polynomial interpolation to get a formula $f$ where $deg(f) = n-1$ if you have have $n$ pairs $(x,f(x))$. Said $f$ will even be unique. But higher degree polynomials could also fit it, infinitely many in fact. And this is all just polynomial functions mind you.
$endgroup$
– Eevee Trainer
2 hours ago
2
2
$begingroup$
You're not guaranteed one unique pattern/function. You can use polynomial interpolation to get a formula $f$ where $deg(f) = n-1$ if you have have $n$ pairs $(x,f(x))$. Said $f$ will even be unique. But higher degree polynomials could also fit it, infinitely many in fact. And this is all just polynomial functions mind you.
$endgroup$
– Eevee Trainer
2 hours ago
$begingroup$
You're not guaranteed one unique pattern/function. You can use polynomial interpolation to get a formula $f$ where $deg(f) = n-1$ if you have have $n$ pairs $(x,f(x))$. Said $f$ will even be unique. But higher degree polynomials could also fit it, infinitely many in fact. And this is all just polynomial functions mind you.
$endgroup$
– Eevee Trainer
2 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The process is called "interpolation". There are an indefinite number of ways to do this for a given set of points.
One way is Lagrange Interpolation which I discuss in the answer linked below :
Explanation of Lagrange Interpolating Polynomial
I have plotted the Lagrange Polynomial for your curve on Desmos. You can see that it matches the pattern you gave. You can interact with the plot by clicking on the link below.
https://www.desmos.com/calculator/enavzwsl09
answered 2 hours ago
SpencerSpencer
9,23722357
$endgroup$
1
$begingroup$
Thanks a bunch! this is very helpful
$endgroup$
– Mohammad AlSaqqa
2 hours ago
$begingroup$
I'm glad it was helpful :). I noticed that you were having issues implementing the Lagrange Interpolating polynomial provided by J. W. Tanner, so I implemented it my self in Desmos. Hopefully that can clear things up for you.
$endgroup$
– Spencer
1 hour ago
add a comment |
$begingroup$
Here's one (not simplified):
$f(x)=dfrac{(x-32)(x-64)(x-128)}{(8-32)(8-64)(8-128)}1+dfrac{(x-8)(x-64)(x-128)}{(32-8)(32-64)(32-128)}8$
$;+dfrac{(x-8)(x-32)(x-128)}{(64-8)(64-32)(64-128)}36+dfrac{(x-8)(x-32)(x-64)}{(128-8)(128-32)(128-64)}98$
answered 2 hours ago
J. W. TannerJ. W. Tanner
6,1311521
$endgroup$
$begingroup$
this equation gave back 68687616x^3-6932668416x^2+186621886464x-1080117166080 which did not fit the curve
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
I had a typo. ($28$ instead of $128$), which I just corrected
$endgroup$
– J. W. Tanner
1 hour ago
$begingroup$
It still does not fit the curve, is my pattern maybe can't be represented as a polynomial?
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
There are many polynomials that fit the four points you gave; I showed how to get one of them
$endgroup$
– J. W. Tanner
27 mins ago
add a comment |
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2 Answers
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active
oldest
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2 Answers
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active
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$begingroup$
The process is called "interpolation". There are an indefinite number of ways to do this for a given set of points.
One way is Lagrange Interpolation which I discuss in the answer linked below :
Explanation of Lagrange Interpolating Polynomial
I have plotted the Lagrange Polynomial for your curve on Desmos. You can see that it matches the pattern you gave. You can interact with the plot by clicking on the link below.
https://www.desmos.com/calculator/enavzwsl09
answered 2 hours ago
SpencerSpencer
9,23722357
$endgroup$
1
$begingroup$
Thanks a bunch! this is very helpful
$endgroup$
– Mohammad AlSaqqa
2 hours ago
$begingroup$
I'm glad it was helpful :). I noticed that you were having issues implementing the Lagrange Interpolating polynomial provided by J. W. Tanner, so I implemented it my self in Desmos. Hopefully that can clear things up for you.
$endgroup$
– Spencer
1 hour ago
add a comment |
$begingroup$
The process is called "interpolation". There are an indefinite number of ways to do this for a given set of points.
One way is Lagrange Interpolation which I discuss in the answer linked below :
Explanation of Lagrange Interpolating Polynomial
I have plotted the Lagrange Polynomial for your curve on Desmos. You can see that it matches the pattern you gave. You can interact with the plot by clicking on the link below.
https://www.desmos.com/calculator/enavzwsl09
answered 2 hours ago
SpencerSpencer
9,23722357
$endgroup$
1
$begingroup$
Thanks a bunch! this is very helpful
$endgroup$
– Mohammad AlSaqqa
2 hours ago
$begingroup$
I'm glad it was helpful :). I noticed that you were having issues implementing the Lagrange Interpolating polynomial provided by J. W. Tanner, so I implemented it my self in Desmos. Hopefully that can clear things up for you.
$endgroup$
– Spencer
1 hour ago
add a comment |
$begingroup$
The process is called "interpolation". There are an indefinite number of ways to do this for a given set of points.
One way is Lagrange Interpolation which I discuss in the answer linked below :
Explanation of Lagrange Interpolating Polynomial
I have plotted the Lagrange Polynomial for your curve on Desmos. You can see that it matches the pattern you gave. You can interact with the plot by clicking on the link below.
https://www.desmos.com/calculator/enavzwsl09
answered 2 hours ago
SpencerSpencer
9,23722357
$endgroup$
The process is called "interpolation". There are an indefinite number of ways to do this for a given set of points.
One way is Lagrange Interpolation which I discuss in the answer linked below :
Explanation of Lagrange Interpolating Polynomial
I have plotted the Lagrange Polynomial for your curve on Desmos. You can see that it matches the pattern you gave. You can interact with the plot by clicking on the link below.
https://www.desmos.com/calculator/enavzwsl09
answered 2 hours ago
SpencerSpencer
9,23722357
edited 1 hour ago
answered 2 hours ago
SpencerSpencer
9,23722357
answered 2 hours ago
SpencerSpencer
9,23722357
answered 2 hours ago
SpencerSpencer
9,23722357
9,23722357
1
$begingroup$
Thanks a bunch! this is very helpful
$endgroup$
– Mohammad AlSaqqa
2 hours ago
$begingroup$
I'm glad it was helpful :). I noticed that you were having issues implementing the Lagrange Interpolating polynomial provided by J. W. Tanner, so I implemented it my self in Desmos. Hopefully that can clear things up for you.
$endgroup$
– Spencer
1 hour ago
add a comment |
1
$begingroup$
Thanks a bunch! this is very helpful
$endgroup$
– Mohammad AlSaqqa
2 hours ago
$begingroup$
I'm glad it was helpful :). I noticed that you were having issues implementing the Lagrange Interpolating polynomial provided by J. W. Tanner, so I implemented it my self in Desmos. Hopefully that can clear things up for you.
$endgroup$
– Spencer
1 hour ago
1
1
$begingroup$
Thanks a bunch! this is very helpful
$endgroup$
– Mohammad AlSaqqa
2 hours ago
$begingroup$
Thanks a bunch! this is very helpful
$endgroup$
– Mohammad AlSaqqa
2 hours ago
$begingroup$
I'm glad it was helpful :). I noticed that you were having issues implementing the Lagrange Interpolating polynomial provided by J. W. Tanner, so I implemented it my self in Desmos. Hopefully that can clear things up for you.
$endgroup$
– Spencer
1 hour ago
$begingroup$
I'm glad it was helpful :). I noticed that you were having issues implementing the Lagrange Interpolating polynomial provided by J. W. Tanner, so I implemented it my self in Desmos. Hopefully that can clear things up for you.
$endgroup$
– Spencer
1 hour ago
add a comment |
$begingroup$
Here's one (not simplified):
$f(x)=dfrac{(x-32)(x-64)(x-128)}{(8-32)(8-64)(8-128)}1+dfrac{(x-8)(x-64)(x-128)}{(32-8)(32-64)(32-128)}8$
$;+dfrac{(x-8)(x-32)(x-128)}{(64-8)(64-32)(64-128)}36+dfrac{(x-8)(x-32)(x-64)}{(128-8)(128-32)(128-64)}98$
answered 2 hours ago
J. W. TannerJ. W. Tanner
6,1311521
$endgroup$
$begingroup$
this equation gave back 68687616x^3-6932668416x^2+186621886464x-1080117166080 which did not fit the curve
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
I had a typo. ($28$ instead of $128$), which I just corrected
$endgroup$
– J. W. Tanner
1 hour ago
$begingroup$
It still does not fit the curve, is my pattern maybe can't be represented as a polynomial?
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
There are many polynomials that fit the four points you gave; I showed how to get one of them
$endgroup$
– J. W. Tanner
27 mins ago
add a comment |
$begingroup$
Here's one (not simplified):
$f(x)=dfrac{(x-32)(x-64)(x-128)}{(8-32)(8-64)(8-128)}1+dfrac{(x-8)(x-64)(x-128)}{(32-8)(32-64)(32-128)}8$
$;+dfrac{(x-8)(x-32)(x-128)}{(64-8)(64-32)(64-128)}36+dfrac{(x-8)(x-32)(x-64)}{(128-8)(128-32)(128-64)}98$
answered 2 hours ago
J. W. TannerJ. W. Tanner
6,1311521
$endgroup$
$begingroup$
this equation gave back 68687616x^3-6932668416x^2+186621886464x-1080117166080 which did not fit the curve
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
I had a typo. ($28$ instead of $128$), which I just corrected
$endgroup$
– J. W. Tanner
1 hour ago
$begingroup$
It still does not fit the curve, is my pattern maybe can't be represented as a polynomial?
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
There are many polynomials that fit the four points you gave; I showed how to get one of them
$endgroup$
– J. W. Tanner
27 mins ago
add a comment |
$begingroup$
Here's one (not simplified):
$f(x)=dfrac{(x-32)(x-64)(x-128)}{(8-32)(8-64)(8-128)}1+dfrac{(x-8)(x-64)(x-128)}{(32-8)(32-64)(32-128)}8$
$;+dfrac{(x-8)(x-32)(x-128)}{(64-8)(64-32)(64-128)}36+dfrac{(x-8)(x-32)(x-64)}{(128-8)(128-32)(128-64)}98$
answered 2 hours ago
J. W. TannerJ. W. Tanner
6,1311521
$endgroup$
Here's one (not simplified):
$f(x)=dfrac{(x-32)(x-64)(x-128)}{(8-32)(8-64)(8-128)}1+dfrac{(x-8)(x-64)(x-128)}{(32-8)(32-64)(32-128)}8$
$;+dfrac{(x-8)(x-32)(x-128)}{(64-8)(64-32)(64-128)}36+dfrac{(x-8)(x-32)(x-64)}{(128-8)(128-32)(128-64)}98$
answered 2 hours ago
J. W. TannerJ. W. Tanner
6,1311521
edited 1 hour ago
answered 2 hours ago
J. W. TannerJ. W. Tanner
6,1311521
answered 2 hours ago
J. W. TannerJ. W. Tanner
6,1311521
answered 2 hours ago
J. W. TannerJ. W. Tanner
6,1311521
6,1311521
$begingroup$
this equation gave back 68687616x^3-6932668416x^2+186621886464x-1080117166080 which did not fit the curve
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
I had a typo. ($28$ instead of $128$), which I just corrected
$endgroup$
– J. W. Tanner
1 hour ago
$begingroup$
It still does not fit the curve, is my pattern maybe can't be represented as a polynomial?
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
There are many polynomials that fit the four points you gave; I showed how to get one of them
$endgroup$
– J. W. Tanner
27 mins ago
add a comment |
$begingroup$
this equation gave back 68687616x^3-6932668416x^2+186621886464x-1080117166080 which did not fit the curve
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
I had a typo. ($28$ instead of $128$), which I just corrected
$endgroup$
– J. W. Tanner
1 hour ago
$begingroup$
It still does not fit the curve, is my pattern maybe can't be represented as a polynomial?
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
There are many polynomials that fit the four points you gave; I showed how to get one of them
$endgroup$
– J. W. Tanner
27 mins ago
$begingroup$
this equation gave back 68687616x^3-6932668416x^2+186621886464x-1080117166080 which did not fit the curve
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
this equation gave back 68687616x^3-6932668416x^2+186621886464x-1080117166080 which did not fit the curve
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
I had a typo. ($28$ instead of $128$), which I just corrected
$endgroup$
– J. W. Tanner
1 hour ago
$begingroup$
I had a typo. ($28$ instead of $128$), which I just corrected
$endgroup$
– J. W. Tanner
1 hour ago
$begingroup$
It still does not fit the curve, is my pattern maybe can't be represented as a polynomial?
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
It still does not fit the curve, is my pattern maybe can't be represented as a polynomial?
$endgroup$
– Mohammad AlSaqqa
1 hour ago
$begingroup$
There are many polynomials that fit the four points you gave; I showed how to get one of them
$endgroup$
– J. W. Tanner
27 mins ago
$begingroup$
There are many polynomials that fit the four points you gave; I showed how to get one of them
$endgroup$
– J. W. Tanner
27 mins ago
add a comment |
Mohammad AlSaqqa is a new contributor. Be nice, and check out our Code of Conduct.
Mohammad AlSaqqa is a new contributor. Be nice, and check out our Code of Conduct.
Mohammad AlSaqqa is a new contributor. Be nice, and check out our Code of Conduct.
Mohammad AlSaqqa is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
You're not guaranteed one unique pattern/function. You can use polynomial interpolation to get a formula $f$ where $deg(f) = n-1$ if you have have $n$ pairs $(x,f(x))$. Said $f$ will even be unique. But higher degree polynomials could also fit it, infinitely many in fact. And this is all just polynomial functions mind you.
$endgroup$
– Eevee Trainer
2 hours ago