Does this method have any other name according to Numerical Analysis? [closed]
$begingroup$
See the book OPTIMIZATION: Algorithms and Applications by Rajesh Kumar Arora, Page-$44$.
2.3.4 Cubic Polynomial Fit
In this method, the function f(x) to be minimized is approximated by a cubic polynomial $P(x)$ as
$$P(x) = a_0 + a_1x + a_2x^2 + a_3x^3 qquad (2.7)$$
If the function $f(x)$ is evaluated at four
different points, then the polynomial coefficients $a0$, $a1$, $a2$, and $a4$
can be evaluated by solving four simultaneous linear equations.
Alternatively, if the value of the function and its derivatives are
available at two points, the polynomial coefficients can still be
evaluated. Once a polynomial is approximated for the function, the
minimum point can be evaluated using the polynomial coefficients.
The first step in this search method is to bracket the minimum of the
func- tion between two points, x1 and x2, such that the following
conditions hold:
$$ f′(x1)f′(x2) < 0 qquad (2.8) $$
Using the information of
$f(x_1), f′(x_1), f(x_2)$, and $f′(x_2)$, the minimum point of the
approximating cubic polynomial can be given as
I am trying to learn this algorithm. I am searching this algorithm on the Internet but isn't finding anything. I have a feeling that this algorithm probably has other name(s).
Does this method/algorithm have any other name according to Numerical Analysis?
optimization numerical-methods
asked Dec 30 '18 at 21:13
user366312user366312
648520
$endgroup$
closed as off-topic by Henrik, clathratus, Andrew, ancientmathematician, Holo Dec 31 '18 at 8:32
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Henrik, clathratus, Andrew, ancientmathematician, Holo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
See the book OPTIMIZATION: Algorithms and Applications by Rajesh Kumar Arora, Page-$44$.
2.3.4 Cubic Polynomial Fit
In this method, the function f(x) to be minimized is approximated by a cubic polynomial $P(x)$ as
$$P(x) = a_0 + a_1x + a_2x^2 + a_3x^3 qquad (2.7)$$
If the function $f(x)$ is evaluated at four
different points, then the polynomial coefficients $a0$, $a1$, $a2$, and $a4$
can be evaluated by solving four simultaneous linear equations.
Alternatively, if the value of the function and its derivatives are
available at two points, the polynomial coefficients can still be
evaluated. Once a polynomial is approximated for the function, the
minimum point can be evaluated using the polynomial coefficients.
The first step in this search method is to bracket the minimum of the
func- tion between two points, x1 and x2, such that the following
conditions hold:
$$ f′(x1)f′(x2) < 0 qquad (2.8) $$
Using the information of
$f(x_1), f′(x_1), f(x_2)$, and $f′(x_2)$, the minimum point of the
approximating cubic polynomial can be given as
I am trying to learn this algorithm. I am searching this algorithm on the Internet but isn't finding anything. I have a feeling that this algorithm probably has other name(s).
Does this method/algorithm have any other name according to Numerical Analysis?
optimization numerical-methods
asked Dec 30 '18 at 21:13
user366312user366312
648520
$endgroup$
closed as off-topic by Henrik, clathratus, Andrew, ancientmathematician, Holo Dec 31 '18 at 8:32
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Henrik, clathratus, Andrew, ancientmathematician, Holo
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Readers may feel you've cut corners by posting mainly a link and an image. I'm sure your own description of an algorithm would be more compelling to read. Youve been around awhile, so presumably you know about MathJax and $LaTeX$. If not, I'd be happy to point you to more information.
$endgroup$
– hardmath
Dec 30 '18 at 21:23
$begingroup$
Are you asking about the cubic interpolation forms? Or about a method to bracket a minimum? Or about a method to find the polynomial minimum based on a bracket?
$endgroup$
– Klaas van Aarsen
Dec 30 '18 at 22:38
$begingroup$
@IlikeSerena, actually, I don't know. Coz, I don't have any idea about the algorithm.
$endgroup$
– user366312
Dec 30 '18 at 22:50
add a comment |
$begingroup$
See the book OPTIMIZATION: Algorithms and Applications by Rajesh Kumar Arora, Page-$44$.
2.3.4 Cubic Polynomial Fit
In this method, the function f(x) to be minimized is approximated by a cubic polynomial $P(x)$ as
$$P(x) = a_0 + a_1x + a_2x^2 + a_3x^3 qquad (2.7)$$
If the function $f(x)$ is evaluated at four
different points, then the polynomial coefficients $a0$, $a1$, $a2$, and $a4$
can be evaluated by solving four simultaneous linear equations.
Alternatively, if the value of the function and its derivatives are
available at two points, the polynomial coefficients can still be
evaluated. Once a polynomial is approximated for the function, the
minimum point can be evaluated using the polynomial coefficients.
The first step in this search method is to bracket the minimum of the
func- tion between two points, x1 and x2, such that the following
conditions hold:
$$ f′(x1)f′(x2) < 0 qquad (2.8) $$
Using the information of
$f(x_1), f′(x_1), f(x_2)$, and $f′(x_2)$, the minimum point of the
approximating cubic polynomial can be given as
I am trying to learn this algorithm. I am searching this algorithm on the Internet but isn't finding anything. I have a feeling that this algorithm probably has other name(s).
Does this method/algorithm have any other name according to Numerical Analysis?
optimization numerical-methods
asked Dec 30 '18 at 21:13
user366312user366312
648520
$endgroup$
See the book OPTIMIZATION: Algorithms and Applications by Rajesh Kumar Arora, Page-$44$.
2.3.4 Cubic Polynomial Fit
In this method, the function f(x) to be minimized is approximated by a cubic polynomial $P(x)$ as
$$P(x) = a_0 + a_1x + a_2x^2 + a_3x^3 qquad (2.7)$$
If the function $f(x)$ is evaluated at four
different points, then the polynomial coefficients $a0$, $a1$, $a2$, and $a4$
can be evaluated by solving four simultaneous linear equations.
Alternatively, if the value of the function and its derivatives are
available at two points, the polynomial coefficients can still be
evaluated. Once a polynomial is approximated for the function, the
minimum point can be evaluated using the polynomial coefficients.
The first step in this search method is to bracket the minimum of the
func- tion between two points, x1 and x2, such that the following
conditions hold:
$$ f′(x1)f′(x2) < 0 qquad (2.8) $$
Using the information of
$f(x_1), f′(x_1), f(x_2)$, and $f′(x_2)$, the minimum point of the
approximating cubic polynomial can be given as
I am trying to learn this algorithm. I am searching this algorithm on the Internet but isn't finding anything. I have a feeling that this algorithm probably has other name(s).
Does this method/algorithm have any other name according to Numerical Analysis?
optimization numerical-methods
optimization numerical-methods
asked Dec 30 '18 at 21:13
user366312user366312
648520
asked Dec 30 '18 at 21:13
user366312user366312
648520
edited Dec 30 '18 at 21:46
user366312
asked Dec 30 '18 at 21:13
user366312user366312
648520
asked Dec 30 '18 at 21:13
user366312user366312
648520
asked Dec 30 '18 at 21:13
user366312user366312
648520
648520
closed as off-topic by Henrik, clathratus, Andrew, ancientmathematician, Holo Dec 31 '18 at 8:32
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Henrik, clathratus, Andrew, ancientmathematician, Holo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Henrik, clathratus, Andrew, ancientmathematician, Holo Dec 31 '18 at 8:32
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Henrik, clathratus, Andrew, ancientmathematician, Holo
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Readers may feel you've cut corners by posting mainly a link and an image. I'm sure your own description of an algorithm would be more compelling to read. Youve been around awhile, so presumably you know about MathJax and $LaTeX$. If not, I'd be happy to point you to more information.
$endgroup$
– hardmath
Dec 30 '18 at 21:23
$begingroup$
Are you asking about the cubic interpolation forms? Or about a method to bracket a minimum? Or about a method to find the polynomial minimum based on a bracket?
$endgroup$
– Klaas van Aarsen
Dec 30 '18 at 22:38
$begingroup$
@IlikeSerena, actually, I don't know. Coz, I don't have any idea about the algorithm.
$endgroup$
– user366312
Dec 30 '18 at 22:50
add a comment |
$begingroup$
Readers may feel you've cut corners by posting mainly a link and an image. I'm sure your own description of an algorithm would be more compelling to read. Youve been around awhile, so presumably you know about MathJax and $LaTeX$. If not, I'd be happy to point you to more information.
$endgroup$
– hardmath
Dec 30 '18 at 21:23
$begingroup$
Are you asking about the cubic interpolation forms? Or about a method to bracket a minimum? Or about a method to find the polynomial minimum based on a bracket?
$endgroup$
– Klaas van Aarsen
Dec 30 '18 at 22:38
$begingroup$
@IlikeSerena, actually, I don't know. Coz, I don't have any idea about the algorithm.
$endgroup$
– user366312
Dec 30 '18 at 22:50
$begingroup$
Readers may feel you've cut corners by posting mainly a link and an image. I'm sure your own description of an algorithm would be more compelling to read. Youve been around awhile, so presumably you know about MathJax and $LaTeX$. If not, I'd be happy to point you to more information.
$endgroup$
– hardmath
Dec 30 '18 at 21:23
$begingroup$
Readers may feel you've cut corners by posting mainly a link and an image. I'm sure your own description of an algorithm would be more compelling to read. Youve been around awhile, so presumably you know about MathJax and $LaTeX$. If not, I'd be happy to point you to more information.
$endgroup$
– hardmath
Dec 30 '18 at 21:23
$begingroup$
Are you asking about the cubic interpolation forms? Or about a method to bracket a minimum? Or about a method to find the polynomial minimum based on a bracket?
$endgroup$
– Klaas van Aarsen
Dec 30 '18 at 22:38
$begingroup$
Are you asking about the cubic interpolation forms? Or about a method to bracket a minimum? Or about a method to find the polynomial minimum based on a bracket?
$endgroup$
– Klaas van Aarsen
Dec 30 '18 at 22:38
$begingroup$
@IlikeSerena, actually, I don't know. Coz, I don't have any idea about the algorithm.
$endgroup$
– user366312
Dec 30 '18 at 22:50
$begingroup$
@IlikeSerena, actually, I don't know. Coz, I don't have any idea about the algorithm.
$endgroup$
– user366312
Dec 30 '18 at 22:50
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Perhaps cubic interpolation will work?
answered Dec 30 '18 at 21:17
gt6989bgt6989b
36.1k22557
$endgroup$
$begingroup$
I googled for "cubic interpolation optimization" and found this relevant video: youtu.be/W-Rcs26oFMM
$endgroup$
– littleO
Dec 30 '18 at 22:09
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Perhaps cubic interpolation will work?
answered Dec 30 '18 at 21:17
gt6989bgt6989b
36.1k22557
$endgroup$
$begingroup$
I googled for "cubic interpolation optimization" and found this relevant video: youtu.be/W-Rcs26oFMM
$endgroup$
– littleO
Dec 30 '18 at 22:09
add a comment |
$begingroup$
Perhaps cubic interpolation will work?
answered Dec 30 '18 at 21:17
gt6989bgt6989b
36.1k22557
$endgroup$
$begingroup$
I googled for "cubic interpolation optimization" and found this relevant video: youtu.be/W-Rcs26oFMM
$endgroup$
– littleO
Dec 30 '18 at 22:09
add a comment |
$begingroup$
Perhaps cubic interpolation will work?
answered Dec 30 '18 at 21:17
gt6989bgt6989b
36.1k22557
$endgroup$
Perhaps cubic interpolation will work?
answered Dec 30 '18 at 21:17
gt6989bgt6989b
36.1k22557
answered Dec 30 '18 at 21:17
gt6989bgt6989b
36.1k22557
answered Dec 30 '18 at 21:17
gt6989bgt6989b
36.1k22557
answered Dec 30 '18 at 21:17
gt6989bgt6989b
36.1k22557
36.1k22557
$begingroup$
I googled for "cubic interpolation optimization" and found this relevant video: youtu.be/W-Rcs26oFMM
$endgroup$
– littleO
Dec 30 '18 at 22:09
add a comment |
$begingroup$
I googled for "cubic interpolation optimization" and found this relevant video: youtu.be/W-Rcs26oFMM
$endgroup$
– littleO
Dec 30 '18 at 22:09
$begingroup$
I googled for "cubic interpolation optimization" and found this relevant video: youtu.be/W-Rcs26oFMM
$endgroup$
– littleO
Dec 30 '18 at 22:09
$begingroup$
I googled for "cubic interpolation optimization" and found this relevant video: youtu.be/W-Rcs26oFMM
$endgroup$
– littleO
Dec 30 '18 at 22:09
add a comment |
$begingroup$
Readers may feel you've cut corners by posting mainly a link and an image. I'm sure your own description of an algorithm would be more compelling to read. Youve been around awhile, so presumably you know about MathJax and $LaTeX$. If not, I'd be happy to point you to more information.
$endgroup$
– hardmath
Dec 30 '18 at 21:23
$begingroup$
Are you asking about the cubic interpolation forms? Or about a method to bracket a minimum? Or about a method to find the polynomial minimum based on a bracket?
$endgroup$
– Klaas van Aarsen
Dec 30 '18 at 22:38
$begingroup$
@IlikeSerena, actually, I don't know. Coz, I don't have any idea about the algorithm.
$endgroup$
– user366312
Dec 30 '18 at 22:50