[R] how to apply summation sign formula to R

[David Winsemius]

On Aug 26, 2013, at 11:59 AM, Sebastian Hersberger wrote:

> Hi David
>
> Thanks for your help.
> I tried it for a simplified example with vectors instead of matrices.
> Once again the formula:
> C = (Σ(from i=0 to i)  A^i ) x B x (Σ(from i=0 to i) A^i )’
>
> I applied it at follows – but couldn’t figure out what’s  
> missing:
>
> library(expm)
> i <- c(0,1,2,3,4,5,6,7,8,9,10)
> A <- c(1,2,3)
> B <- 5
> res1 <- Reduce("+", lapply(1:i, function(i) A ^ i) )

I thought I suggested the use of the  %^% operator? Doing A^i is only  
going to give you the element-wise values to the i-th power, not the  
matrix to the i-th power. (But perhaps that is what you actually  
wanted, which was never expalined in the portions of the thread which  
I read. And I would have expected A %^% 0 to be I, so it shouldn't  
really matter whether the iteration limit was from 0 to 10 or from 1  
to 10.)

And you might need to use a final sum( around that Reduce()-ed value  
if you were not planning on using matrix multiplication. I thought  
this was supposed to be using matrix math. Please explain the purpose  
of this effort.


> res2 <- res1 * B
>> res3 <- res2 * t(res1)
>> res3
>      [,1] [,2] [,3]
> [1,]   20   45   80
>
>
> My expected result must be in this case a single number, which is  
> the sum of i=0 to i=10:
> i=0 ⇒ 5
> i=1 ⇒ 1*5*1 + 1*5*2+1*5*3 + 2*5*1+2*5*2 + 2*5*3+3*5*1+3*5*2+3*5*3 =  
> 180
> i = 2 ⇒ ........... (=985)
> i=3 ⇒ ....
> ...
> i=10 ⇒ =............
>
>
>
> After having adjusted the above provided „R-code“: what is  
> necessary to get from the „vector-solution“ to the matrix- 
> solution“: would it be enough to:
> -	define A and B as matrices
> -	and replace * by %*%  ?

Perhaps also replacing ^ with %^%. Just a guess. I'm unable to  
decipher from your still sketchy description what the mathematical  
operations are supposed to be or do.

-- 
David.


> From: David Winsemius [dwinsemius at comcast.net]
> Sent: Sunday, August 25, 2013 3:51 AM
> To: David Winsemius
> Cc: Sebastian Hersberger; r-help at r-project.org
> Subject: Re: [R] how to apply summation sign formula to R
>
> On Aug 24, 2013, at 6:43 PM, David Winsemius wrote:
>
>>
>> On Aug 24, 2013, at 2:13 PM, Sebastian Hersberger wrote:
>>
>>> Thanks. I restate my problem/question and hope its better  
>>> understandable now.
>>>
>>> Let us define A and B as kxk matrices. C is the output (matrix),  
>>> which I try to calculate for differnt i values.
>>>
>>> So for example: I want to caluclate the matrix C for the value i=10:
>>>
>>> Therefore, I set:
>>>
>>> i <- c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
>>>
>>> Finally, I have to define the summation formula in R. My question  
>>> is how this following summation formula has to be applied to R.
>>>
>>> The arithmetic form of the formula equals:
>>>
>>> C = (Σ(from i=0 to i)  A^i ) x B x (Σ(from i=0 to i) A^i )’
>>>
>>> Which means:
>>> matrix C equals the sum from i=0 to i times matrix A to the power  
>>> of i
>>
>> That first term might be (untested in absence of test data by  
>> questioner) :
>>
>> require(expm)
>> res1 <- Reduce("+", lapply(1:i, function(i) powm(A, i) ) )
>
> I did test it and both my memory about the functions name and its  
> operation were wrong. Try:
>
> res1 <- Reduce("+", lapply(1:i, function(i) A %^% i) )
>
>>
>>> times matrix B
>>
>> res2 <- res1 %*% B
>>
>>> times the transposed/invers of the sum from i=0 to i times matrix  
>>> A to the power of i
>>>
>> Well, the transpose was not generally the inverse when I took  
>> linear algebra, but as I said I didn't make it to all the sessions.  
>> (I'm assuming it is the transpose.)
>>
>> res3 <- res2 %*% t(res1)
>>
>> You could probably do it all in one line, but I like to check my  
>> itermediates.
>>
>> --
>> David.
>>
>>> I hope I were able to specify my problem in an understandable way.  
>>> If not, please let me know.
>>>
>>> Thanks and regards
>>> Sebastian
>>>
>>> ________________________________________
>>> From: David Winsemius [dwinsemius at comcast.net]
>>> Sent: Saturday, August 24, 2013 8:35 PM
>>> To: Sebastian Hersberger
>>> Cc: r-help at r-project.org
>>> Subject: Re: [R] how to apply summation sign formula to R
>>>
>>> On Aug 24, 2013, at 6:37 AM, Sebastian Hersberger wrote:
>>>
>>>> Hi all
>>>>
>>>> I have a short question relating to the usage of the summation  
>>>> sign in R.
>>>>
>>>> Let's define A and B as two kxk matrice.
>>>> My goal is to calculate the matrix C for the periods from 1 to  
>>>> 200 (n=1-200).
>>>>
>>>> C^(n) = Σ_(j=1)^n [(Σ_(i=1)^(j-1) A^i ) B (Σ_(i=1)^(j-1)  
>>>> A^i)’  ]
>>>>
>>>> How has that to be implemented in R (lets say for example for  
>>>> period = n = 150)?
>>>
>>> I don't follow all this notation but you might want to look at the  
>>> expm package that has a powm function. It's possible that the expm  
>>> function may do what you want, but as I said I was unclear what  
>>> the equation was attempting to do. Perhaps if you explained the  
>>> steps in mathematical language, then other viewers who actually  
>>> woke up to attend their linear algebra course might have a better  
>>> chance of offering assistance.
>>>
>>> ( When I saw the earlier copy yesterday, I had hopes that such  
>>> wiser viewers might chime in and still have such hopes.)
>>>
>>> --
>>>
>>> David Winsemius
>>> Alameda, CA, USA
>>>
>>
>> David Winsemius
>> Alameda, CA, USA
>>
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>> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>
> David Winsemius
> Alameda, CA, USA
>

David Winsemius, MD
Alameda, CA, USA