Hi, I want find all roots for the following polynomial :
a <- c(-0.07, 0.17); b <- c(1, -4); cc <- matrix(c(0.24, 0.00, -0.08,
-0.31), 2); d <- matrix(c(0, 0, -0.13, -0.37), 2); e <- matrix(c(0.2, 0,
-0.06, -0.34), 2)
A1 <- diag(2) + a %*% t(b) + cc; A2 <- -cc + d; A3 <- -d + e; A4 <-
-e
fn <- function(z)
{
y <- diag(2) - A1*z - A2*z^2 - A3*z^3 - A4*z^4
return(det(y))
}; uniroot(fn, c(-10, 1))
Using uniroot function, I got only one solution of that. Is there any
function to get all four solutions? I looked at polyroot() function, but I
do not think it will work for my problem, because, my coef. are matrix, nor
number
Thanks
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On Sun, Jan 11, 2009 at 6:03 AM, RON70 <ron_michael70 at yahoo.com> wrote:> > Hi, I want find all roots for the following polynomial : > > a <- c(-0.07, 0.17); b <- c(1, -4); cc <- matrix(c(0.24, 0.00, -0.08, > -0.31), 2); d <- matrix(c(0, 0, -0.13, -0.37), 2); e <- matrix(c(0.2, 0, > -0.06, -0.34), 2) > A1 <- diag(2) + a %*% t(b) + cc; A2 <- -cc + d; A3 <- -d + e; A4 <- -e > fn <- function(z) > { > y <- diag(2) - A1*z - A2*z^2 - A3*z^3 - A4*z^4 > return(det(y)) > }; uniroot(fn, c(-10, 1)) > > Using uniroot function, I got only one solution of that. Is there any > function to get all four solutions? I looked at polyroot() function, but I > do not think it will work for my problem, because, my coef. are matrix, nor > numberUse curve to plot the curve of your function. Then, see where the roots are, and use uniroot with a small interval around the roots to determine their exact value. Example: f <- function(x) x^2-1 curve(f,-5,5) uniroot(f,c(-2,-0.2)) uniroot(f,c(0.2,2)) Paul
In theory,
you could define the following 2 functions
powermat <- function(myvec) {
powervec <- function(x,maxpower)
sapply(0:maxpower,function(n)x^n)
sapply(myvec,function(x)powervec(x,length(myvec)-1))
}
polycoeffs <- function(fn,order,support=0:order)
solve(t(powermat(support)),sapply(support,Vectorize(fn)))
and then use polycoeffs(fn,4)
with your function fn to extract the coefficients of the polynomial.
polycoeffs takes a function and the order of a polynomial
as its input and computes the coefficients of the polynomial
or the given order with the same values as the function
at the points 0:order.
If the function is a polynomial of the given order, it gives you
exactly the coefficients you need.
You may use another set of support points (points where the function is
evaluated) as an optional argument.
Still, this method is not extremely reliable. If the support points are
not chosen well, you might get rather unreliable results.
A safer way would be to use a computer algebra system to extract the
coefficients of your polynomial. You could use Ryacas to do this from R.
But you still would have to grasp the syntax of yacas.
Paul Smith wrote:> On Sun, Jan 11, 2009 at 6:03 AM, RON70 <ron_michael70 at yahoo.com>
wrote:
>> Hi, I want find all roots for the following polynomial :
>>
>> a <- c(-0.07, 0.17); b <- c(1, -4); cc <- matrix(c(0.24, 0.00,
-0.08,
>> -0.31), 2); d <- matrix(c(0, 0, -0.13, -0.37), 2); e <-
matrix(c(0.2, 0,
>> -0.06, -0.34), 2)
>> A1 <- diag(2) + a %*% t(b) + cc; A2 <- -cc + d; A3 <- -d + e;
A4 <- -e
>> fn <- function(z)
>> {
>> y <- diag(2) - A1*z - A2*z^2 - A3*z^3 - A4*z^4
>> return(det(y))
>> }; uniroot(fn, c(-10, 1))
>>
>> Using uniroot function, I got only one solution of that. Is there any
>> function to get all four solutions? I looked at polyroot() function,
but I
>> do not think it will work for my problem, because, my coef. are matrix,
nor
>> number
>
> Use curve to plot the curve of your function. Then, see where the
> roots are, and use uniroot with a small interval around the roots to
> determine their exact value.
>
> Example:
>
> f <- function(x) x^2-1
> curve(f,-5,5)
> uniroot(f,c(-2,-0.2))
> uniroot(f,c(0.2,2))
>
> Paul
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>
>
> ------------------------------------------------------------------------
>
>
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> Checked by AVG - http://www.avg.com
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6:01 PM
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Faculty of Computer Science
Computer Supported Didactics Working Group
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Hi,
You can use the "polynomial" package to solve your problem.
The key step is to find the exact polynomial representation of fn(). Noting
that it is a 8-th degree polynomial, we can get its exact form using the
poly.calc() function. Once we have that, it is a simple matter of finding the
roots using the solve() function.
require(polynomial)
a <- c(-0.07, 0.17)
b <- c(1, -4)
cc <- matrix(c(0.24, 0.00, -0.08, -0.31), 2)
d <- matrix(c(0, 0, -0.13, -0.37), 2)
e <- matrix(c(0.2, 0, -0.06, -0.34), 2)
A1 <- diag(2) + a %*% t(b) + cc
A2 <- -cc + d
A3 <- -d + e
A4 <- -e
# I am vectorizing your function
fn <- function(z)
{
sapply(z, function(z) {
y <- diag(2) - A1*z - A2*z^2 - A3*z^3 - A4*z^4
det(y)
})
}
x <- seq(-5, 5, length=9) # note we need 9 points to exactly determine a 8-th
degree polynomial
y <- fn(x)
p <- poly.calc(x, y) # uses Lagrange interpolation to determine polynomial
form
p> 1 - 1.18*x + 0.2777*x^2 - 0.2941*x^3 - 0.1004*x^4 + 0.3664*x^5 - 0.0636*x^6
+ 0.062*x^7 - 0.068*x^8
# plot showing that p is the exact polynomial representation of fn(z)
pfunc <- as.function(p)
x1 <-seq(-5, 5, length=100)
plot(x1, fn(x1),type="l")
lines(x1, pfunc(x1), col=2, lty=2)
solve(p) # gives you the roots (some are, of course, complex)
Hope this helps,
Ravi.
____________________________________________________________________
Ravi Varadhan, Ph.D.
Assistant Professor,
Division of Geriatric Medicine and Gerontology
School of Medicine
Johns Hopkins University
Ph. (410) 502-2619
email: rvaradhan at jhmi.edu
----- Original Message -----
From: RON70 <ron_michael70 at yahoo.com>
Date: Sunday, January 11, 2009 1:05 am
Subject: [R] How to get solution of following polynomial?
To: r-help at r-project.org
> Hi, I want find all roots for the following polynomial :
>
> a <- c(-0.07, 0.17); b <- c(1, -4); cc <- matrix(c(0.24, 0.00,
-0.08,
> -0.31), 2); d <- matrix(c(0, 0, -0.13, -0.37), 2); e <-
matrix(c(0.2,
> 0,
> -0.06, -0.34), 2)
> A1 <- diag(2) + a %*% t(b) + cc; A2 <- -cc + d; A3 <- -d + e; A4
<- -e
> fn <- function(z)
> {
> y <- diag(2) - A1*z - A2*z^2 - A3*z^3 - A4*z^4
> return(det(y))
> }; uniroot(fn, c(-10, 1))
>
> Using uniroot function, I got only one solution of that. Is there any
> function to get all four solutions? I looked at polyroot() function,
> but I
> do not think it will work for my problem, because, my coef. are
> matrix, nor
> number
>
> Thanks
>
>
>
> --
> View this message in context:
> Sent from the R help mailing list archive at Nabble.com.
>
> ______________________________________________
> R-help at r-project.org mailing list
>
> PLEASE do read the posting guide
> and provide commented, minimal, self-contained, reproducible code.
Hi Ravi, Thanks for this reply. However I could not understand meaning of "vectorizing the function". Can you please be little bit elaborate on that? Secondly the package "polynomial" is not available in CRAN it seems. What is the alternate package? Thanks, Ravi Varadhan wrote:> > Hi, > > You can use the "polynomial" package to solve your problem. > > The key step is to find the exact polynomial representation of fn(). > Noting that it is a 8-th degree polynomial, we can get its exact form > using the poly.calc() function. Once we have that, it is a simple matter > of finding the roots using the solve() function. > > require(polynomial) > > a <- c(-0.07, 0.17) > b <- c(1, -4) > cc <- matrix(c(0.24, 0.00, -0.08, -0.31), 2) > d <- matrix(c(0, 0, -0.13, -0.37), 2) > e <- matrix(c(0.2, 0, -0.06, -0.34), 2) > A1 <- diag(2) + a %*% t(b) + cc > A2 <- -cc + d > A3 <- -d + e > A4 <- -e > > # I am vectorizing your function > fn <- function(z) > { > sapply(z, function(z) { > y <- diag(2) - A1*z - A2*z^2 - A3*z^3 - A4*z^4 > det(y) > }) > } > > > x <- seq(-5, 5, length=9) # note we need 9 points to exactly determine a > 8-th degree polynomial > y <- fn(x) > > p <- poly.calc(x, y) # uses Lagrange interpolation to determine > polynomial form > p >> 1 - 1.18*x + 0.2777*x^2 - 0.2941*x^3 - 0.1004*x^4 + 0.3664*x^5 - >> 0.0636*x^6 + 0.062*x^7 - 0.068*x^8 > > # plot showing that p is the exact polynomial representation of fn(z) > pfunc <- as.function(p) > x1 <-seq(-5, 5, length=100) > plot(x1, fn(x1),type="l") > lines(x1, pfunc(x1), col=2, lty=2) > > solve(p) # gives you the roots (some are, of course, complex) > > > Hope this helps, > Ravi. > > > > ____________________________________________________________________ > > Ravi Varadhan, Ph.D. > Assistant Professor, > Division of Geriatric Medicine and Gerontology > School of Medicine > Johns Hopkins University > > Ph. (410) 502-2619 > email: rvaradhan at jhmi.edu > > > ----- Original Message ----- > From: RON70 <ron_michael70 at yahoo.com> > Date: Sunday, January 11, 2009 1:05 am > Subject: [R] How to get solution of following polynomial? > To: r-help at r-project.org > > >> Hi, I want find all roots for the following polynomial : >> >> a <- c(-0.07, 0.17); b <- c(1, -4); cc <- matrix(c(0.24, 0.00, -0.08, >> -0.31), 2); d <- matrix(c(0, 0, -0.13, -0.37), 2); e <- matrix(c(0.2, >> 0, >> -0.06, -0.34), 2) >> A1 <- diag(2) + a %*% t(b) + cc; A2 <- -cc + d; A3 <- -d + e; A4 <- -e >> fn <- function(z) >> { >> y <- diag(2) - A1*z - A2*z^2 - A3*z^3 - A4*z^4 >> return(det(y)) >> }; uniroot(fn, c(-10, 1)) >> >> Using uniroot function, I got only one solution of that. Is there any >> function to get all four solutions? I looked at polyroot() function, >> but I >> do not think it will work for my problem, because, my coef. are >> matrix, nor >> number >> >> Thanks >> >> >> >> -- >> View this message in context: >> Sent from the R help mailing list archive at Nabble.com. >> >> ______________________________________________ >> R-help at r-project.org mailing list >> >> PLEASE do read the posting guide >> and provide commented, minimal, self-contained, reproducible code. > > ______________________________________________ > R-help at r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide > http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. > >-- View this message in context: http://www.nabble.com/How-to-get-solution-of-following-polynomial--tp21396441p21406350.html Sent from the R help mailing list archive at Nabble.com.