MaplePrimes Questions

In Hansen's book "A table of Series and Products" there is an unsourced listed identity 43.6.1:

S1 := Sum(2^k*tanh(2^k*x), k = 0 .. n - 1) = 2^n*coth(2^n*x) -   coth(x)

It appears to be to be numerically correct, and can be obtained from a second identity

S6:=Sum(2^(-k)*tanh(2^(-k)*x), k = 1 .. n) =- 2^(-n)*coth(2^(-n)*x) + coth(x)

by "summing in reverse order". S6 is also numerically correct.

Starting from S1 and reversing the order, gives

S2 := Sum(2^(n - k - 1)*tanh(2^(n - k - 1)*x), k = 0 .. n - 1) = 2^n*coth(2^n*x) - coth(x)

Then shifting the index gives

 S3 := Sum(2^(-k)*tanh(2^(-k)*x), k = -n + 1 .. 0) =  2^n*coth(2^n*x) - coth(x)

both of which test OK numerically since n>0. Now it begins to get strange.

In S3, let n->-n (highly illegal) to obtain

S4 := Sum(2^(-k)*tanh(2^(-k)*x), k = n + 1 .. 0) = 2^(-n)*coth(2^(-n)*x) - coth(x)

and, surprisingly Maple tests S4 to be correct numerically.

Here's how that happened. The lower limit of the S4 sum exceeds the upper limit, so there are three possible ways to interpret this:

1. In standard mathematical usage, this is usually set to zero by caveat;

2. In a practical sense, one might expect the order of summation to be irrelevant so S4 is the same as

S5:= Sum(2^(-k)*tanh(2^(-k)*x), k = 0 .. n + 1) = 2^(-n)*coth(2^(-n)*x) - coth(x)

but that fails numerically.

3. Maple has a built-in algorithm (see the help page on "Sum") that interprets

Sum(f(k), k = m .. n) = -Sum(f(k), k = n + 1 .. m - 1)

when m exceeds n and that is why S4 works numerically and, when applied to S4 gives the result S6.

So where did Maple's prescription come from, and what is it's justification?

Thank you

This is from a graph G with vertex set {0,1,2,3,4,5,6,7,8,9} always labelled from {0,1,2,3,...,n-1}

L:=[{{0, 1}, {0, 3}, {0, 5}, {0, 7}, {2, 9}}, {{0, 1}, {0, 3}, {0, 5}, {0, 7}, {4, 9}}, {{0, 1}, {0, 3}, {0, 5}, {0, 7}, {6, 9}}, {{0, 1}, {0, 3}, {0, 5}, {0, 7}, {8, 9}}, {{0, 1}, {0, 3}, {0, 5}, {0, 9}, {2, 7}}, {{0, 1}, {0, 3}, {0, 5}, {0, 9}, {4, 7}}, {{0, 1}, {0, 3}, {0, 5}, {0, 9}, {6, 7}}, {{0, 1}, {0, 3}, {0, 5}, {0, 9}, {7, 8}}, {{0, 1}, {0, 3}, {0, 7}, {0, 9}, {2, 5}}, {{0, 1}, {0, 3}, {0, 7}, {0, 9}, {4, 5}}, {{0, 1}, {0, 3}, {0, 7}, {0, 9}, {5, 6}}, {{0, 1}, {0, 3}, {0, 7}, {0, 9}, {5, 8}}, {{0, 1}, {0, 5}, {0, 7}, {0, 9}, {2, 3}}, {{0, 1}, {0, 5}, {0, 7}, {0, 9}, {3, 4}}, {{0, 1}, {0, 5}, {0, 7}, {0, 9}, {3, 6}}, {{0, 1}, {0, 5}, {0, 7}, {0, 9}, {3, 8}}, {{0, 1}, {1, 2}, {1, 4}, {1, 6}, {3, 8}}, {{0, 1}, {1, 2}, {1, 4}, {1, 6}, {5, 8}}, {{0, 1}, {1, 2}, {1, 4}, {1, 6}, {7, 8}}, {{0, 1}, {1, 2}, {1, 4}, {1, 6}, {8, 9}}, {{0, 1}, {1, 2}, {1, 4}, {1, 8}, {3, 6}}, {{0, 1}, {1, 2}, {1, 4}, {1, 8}, {5, 6}}, {{0, 1}, {1, 2}, {1, 4}, {1, 8}, {6, 7}}, {{0, 1}, {1, 2}, {1, 4}, {1, 8}, {6, 9}}, {{0, 1}, {1, 2}, {1, 6}, {1, 8}, {3, 4}}, {{0, 1}, {1, 2}, {1, 6}, {1, 8}, {4, 5}}, {{0, 1}, {1, 2}, {1, 6}, {1, 8}, {4, 7}}, {{0, 1}, {1, 2}, {1, 6}, {1, 8}, {4, 9}}, {{0, 1}, {1, 4}, {1, 6}, {1, 8}, {2, 3}}, {{0, 1}, {1, 4}, {1, 6}, {1, 8}, {2, 5}}, {{0, 1}, {1, 4}, {1, 6}, {1, 8}, {2, 7}}, {{0, 1}, {1, 4}, {1, 6}, {1, 8}, {2, 9}}, {{0, 1}, {2, 3}, {2, 5}, {2, 7}, {2, 9}}, {{0, 1}, {2, 3}, {3, 4}, {3, 6}, {3, 8}}, {{0, 1}, {2, 5}, {4, 5}, {5, 6}, {5, 8}}, {{0, 1}, {2, 7}, {4, 7}, {6, 7}, {7, 8}}, {{0, 1}, {2, 9}, {4, 9}, {6, 9}, {8, 9}}, {{0, 1}, {3, 4}, {4, 5}, {4, 7}, {4, 9}}, {{0, 1}, {3, 6}, {5, 6}, {6, 7}, {6, 9}}, {{0, 1}, {3, 8}, {5, 8}, {7, 8}, {8, 9}}, {{0, 3}, {0, 5}, {0, 7}, {0, 9}, {1, 2}}, {{0, 3}, {0, 5}, {0, 7}, {0, 9}, {1, 4}}, {{0, 3}, {0, 5}, {0, 7}, {0, 9}, {1, 6}}, {{0, 3}, {0, 5}, {0, 7}, {0, 9}, {1, 8}}, {{0, 3}, {1, 2}, {1, 4}, {1, 6}, {1, 8}}, {{0, 3}, {1, 2}, {2, 5}, {2, 7}, {2, 9}}, {{0, 3}, {1, 2}, {3, 4}, {3, 6}, {3, 8}}, {{0, 3}, {1, 4}, {2, 3}, {3, 6}, {3, 8}}, {{0, 3}, {1, 4}, {4, 5}, {4, 7}, {4, 9}}, {{0, 3}, {1, 6}, {2, 3}, {3, 4}, {3, 8}}, {{0, 3}, {1, 6}, {5, 6}, {6, 7}, {6, 9}}, {{0, 3}, {1, 8}, {2, 3}, {3, 4}, {3, 6}}, {{0, 3}, {1, 8}, {5, 8}, {7, 8}, {8, 9}}, {{0, 3}, {2, 3}, {3, 4}, {3, 6}, {5, 8}}, {{0, 3}, {2, 3}, {3, 4}, {3, 6}, {7, 8}}, {{0, 3}, {2, 3}, {3, 4}, {3, 6}, {8, 9}}, {{0, 3}, {2, 3}, {3, 4}, {3, 8}, {5, 6}}, {{0, 3}, {2, 3}, {3, 4}, {3, 8}, {6, 7}}, {{0, 3}, {2, 3}, {3, 4}, {3, 8}, {6, 9}}, {{0, 3}, {2, 3}, {3, 6}, {3, 8}, {4, 5}}, {{0, 3}, {2, 3}, {3, 6}, {3, 8}, {4, 7}}, {{0, 3}, {2, 3}, {3, 6}, {3, 8}, {4, 9}}, {{0, 3}, {2, 5}, {3, 4}, {3, 6}, {3, 8}}, {{0, 3}, {2, 5}, {4, 5}, {5, 6}, {5, 8}}, {{0, 3}, {2, 7}, {3, 4}, {3, 6}, {3, 8}}, {{0, 3}, {2, 7}, {4, 7}, {6, 7}, {7, 8}}, {{0, 3}, {2, 9}, {3, 4}, {3, 6}, {3, 8}}, {{0, 3}, {2, 9}, {4, 9}, {6, 9}, {8, 9}}, {{0, 5}, {1, 2}, {1, 4}, {1, 6}, {1, 8}}, {{0, 5}, {1, 2}, {2, 3}, {2, 7}, {2, 9}}, {{0, 5}, {1, 2}, {4, 5}, {5, 6}, {5, 8}}, {{0, 5}, {1, 4}, {2, 5}, {5, 6}, {5, 8}}, {{0, 5}, {1, 4}, {3, 4}, {4, 7}, {4, 9}}, {{0, 5}, {1, 6}, {2, 5}, {4, 5}, {5, 8}}, {{0, 5}, {1, 6}, {3, 6}, {6, 7}, {6, 9}}, {{0, 5}, {1, 8}, {2, 5}, {4, 5}, {5, 6}}, {{0, 5}, {1, 8}, {3, 8}, {7, 8}, {8, 9}}, {{0, 5}, {2, 3}, {3, 4}, {3, 6}, {3, 8}}, {{0, 5}, {2, 3}, {4, 5}, {5, 6}, {5, 8}}, {{0, 5}, {2, 5}, {3, 4}, {5, 6}, {5, 8}}, {{0, 5}, {2, 5}, {3, 6}, {4, 5}, {5, 8}}, {{0, 5}, {2, 5}, {3, 8}, {4, 5}, {5, 6}}, {{0, 5}, {2, 5}, {4, 5}, {5, 6}, {7, 8}}, {{0, 5}, {2, 5}, {4, 5}, {5, 6}, {8, 9}}, {{0, 5}, {2, 5}, {4, 5}, {5, 8}, {6, 7}}, {{0, 5}, {2, 5}, {4, 5}, {5, 8}, {6, 9}}, {{0, 5}, {2, 5}, {4, 7}, {5, 6}, {5, 8}}, {{0, 5}, {2, 5}, {4, 9}, {5, 6}, {5, 8}}, {{0, 5}, {2, 7}, {4, 5}, {5, 6}, {5, 8}}, {{0, 5}, {2, 7}, {4, 7}, {6, 7}, {7, 8}}, {{0, 5}, {2, 9}, {4, 5}, {5, 6}, {5, 8}}, {{0, 5}, {2, 9}, {4, 9}, {6, 9}, {8, 9}}, {{0, 7}, {1, 2}, {1, 4}, {1, 6}, {1, 8}}, {{0, 7}, {1, 2}, {2, 3}, {2, 5}, {2, 9}}, {{0, 7}, {1, 2}, {4, 7}, {6, 7}, {7, 8}}, {{0, 7}, {1, 4}, {2, 7}, {6, 7}, {7, 8}}, {{0, 7}, {1, 4}, {3, 4}, {4, 5}, {4, 9}}, {{0, 7}, {1, 6}, {2, 7}, {4, 7}, {7, 8}}, {{0, 7}, {1, 6}, {3, 6}, {5, 6}, {6, 9}}, {{0, 7}, {1, 8}, {2, 7}, {4, 7}, {6, 7}}, {{0, 7}, {1, 8}, {3, 8}, {5, 8}, {8, 9}}, {{0, 7}, {2, 3}, {3, 4}, {3, 6}, {3, 8}}, {{0, 7}, {2, 3}, {4, 7}, {6, 7}, {7, 8}}, {{0, 7}, {2, 5}, {4, 5}, {5, 6}, {5, 8}}, {{0, 7}, {2, 5}, {4, 7}, {6, 7}, {7, 8}}, {{0, 7}, {2, 7}, {3, 4}, {6, 7}, {7, 8}}, {{0, 7}, {2, 7}, {3, 6}, {4, 7}, {7, 8}}, {{0, 7}, {2, 7}, {3, 8}, {4, 7}, {6, 7}}, {{0, 7}, {2, 7}, {4, 5}, {6, 7}, {7, 8}}, {{0, 7}, {2, 7}, {4, 7}, {5, 6}, {7, 8}}, {{0, 7}, {2, 7}, {4, 7}, {5, 8}, {6, 7}}, {{0, 7}, {2, 7}, {4, 7}, {6, 7}, {8, 9}}, {{0, 7}, {2, 7}, {4, 7}, {6, 9}, {7, 8}}, {{0, 7}, {2, 7}, {4, 9}, {6, 7}, {7, 8}}, {{0, 7}, {2, 9}, {4, 7}, {6, 7}, {7, 8}}, {{0, 7}, {2, 9}, {4, 9}, {6, 9}, {8, 9}}, {{0, 9}, {1, 2}, {1, 4}, {1, 6}, {1, 8}}, {{0, 9}, {1, 2}, {2, 3}, {2, 5}, {2, 7}}, {{0, 9}, {1, 2}, {4, 9}, {6, 9}, {8, 9}}, {{0, 9}, {1, 4}, {2, 9}, {6, 9}, {8, 9}}, {{0, 9}, {1, 4}, {3, 4}, {4, 5}, {4, 7}}, {{0, 9}, {1, 6}, {2, 9}, {4, 9}, {8, 9}}, {{0, 9}, {1, 6}, {3, 6}, {5, 6}, {6, 7}}, {{0, 9}, {1, 8}, {2, 9}, {4, 9}, {6, 9}}, {{0, 9}, {1, 8}, {3, 8}, {5, 8}, {7, 8}}, {{0, 9}, {2, 3}, {3, 4}, {3, 6}, {3, 8}}, {{0, 9}, {2, 3}, {4, 9}, {6, 9}, {8, 9}}, {{0, 9}, {2, 5}, {4, 5}, {5, 6}, {5, 8}}, {{0, 9}, {2, 5}, {4, 9}, {6, 9}, {8, 9}}, {{0, 9}, {2, 7}, {4, 7}, {6, 7}, {7, 8}}, {{0, 9}, {2, 7}, {4, 9}, {6, 9}, {8, 9}}, {{0, 9}, {2, 9}, {3, 4}, {6, 9}, {8, 9}}, {{0, 9}, {2, 9}, {3, 6}, {4, 9}, {8, 9}}, {{0, 9}, {2, 9}, {3, 8}, {4, 9}, {6, 9}}, {{0, 9}, {2, 9}, {4, 5}, {6, 9}, {8, 9}}, {{0, 9}, {2, 9}, {4, 7}, {6, 9}, {8, 9}}, {{0, 9}, {2, 9}, {4, 9}, {5, 6}, {8, 9}}, {{0, 9}, {2, 9}, {4, 9}, {5, 8}, {6, 9}}, {{0, 9}, {2, 9}, {4, 9}, {6, 7}, {8, 9}}, {{0, 9}, {2, 9}, {4, 9}, {6, 9}, {7, 8}}, {{1, 2}, {2, 3}, {2, 5}, {2, 7}, {4, 9}}, {{1, 2}, {2, 3}, {2, 5}, {2, 7}, {6, 9}}, {{1, 2}, {2, 3}, {2, 5}, {2, 7}, {8, 9}}, {{1, 2}, {2, 3}, {2, 5}, {2, 9}, {4, 7}}, {{1, 2}, {2, 3}, {2, 5}, {2, 9}, {6, 7}}, {{1, 2}, {2, 3}, {2, 5}, {2, 9}, {7, 8}}, {{1, 2}, {2, 3}, {2, 7}, {2, 9}, {4, 5}}, {{1, 2}, {2, 3}, {2, 7}, {2, 9}, {5, 6}}, {{1, 2}, {2, 3}, {2, 7}, {2, 9}, {5, 8}}, {{1, 2}, {2, 5}, {2, 7}, {2, 9}, {3, 4}}, {{1, 2}, {2, 5}, {2, 7}, {2, 9}, {3, 6}}, {{1, 2}, {2, 5}, {2, 7}, {2, 9}, {3, 8}}, {{1, 2}, {3, 4}, {4, 5}, {4, 7}, {4, 9}}, {{1, 2}, {3, 6}, {5, 6}, {6, 7}, {6, 9}}, {{1, 2}, {3, 8}, {5, 8}, {7, 8}, {8, 9}}, {{1, 4}, {2, 3}, {2, 5}, {2, 7}, {2, 9}}, {{1, 4}, {2, 3}, {4, 5}, {4, 7}, {4, 9}}, {{1, 4}, {2, 5}, {3, 4}, {4, 7}, {4, 9}}, {{1, 4}, {2, 7}, {3, 4}, {4, 5}, {4, 9}}, {{1, 4}, {2, 9}, {3, 4}, {4, 5}, {4, 7}}, {{1, 4}, {3, 4}, {4, 5}, {4, 7}, {6, 9}}, {{1, 4}, {3, 4}, {4, 5}, {4, 7}, {8, 9}}, {{1, 4}, {3, 4}, {4, 5}, {4, 9}, {6, 7}}, {{1, 4}, {3, 4}, {4, 5}, {4, 9}, {7, 8}}, {{1, 4}, {3, 4}, {4, 7}, {4, 9}, {5, 6}}, {{1, 4}, {3, 4}, {4, 7}, {4, 9}, {5, 8}}, {{1, 4}, {3, 6}, {4, 5}, {4, 7}, {4, 9}}, {{1, 4}, {3, 6}, {5, 6}, {6, 7}, {6, 9}}, {{1, 4}, {3, 8}, {4, 5}, {4, 7}, {4, 9}}, {{1, 4}, {3, 8}, {5, 8}, {7, 8}, {8, 9}}, {{1, 6}, {2, 3}, {2, 5}, {2, 7}, {2, 9}}, {{1, 6}, {2, 3}, {5, 6}, {6, 7}, {6, 9}}, {{1, 6}, {2, 5}, {3, 6}, {6, 7}, {6, 9}}, {{1, 6}, {2, 7}, {3, 6}, {5, 6}, {6, 9}}, {{1, 6}, {2, 9}, {3, 6}, {5, 6}, {6, 7}}, {{1, 6}, {3, 4}, {4, 5}, {4, 7}, {4, 9}}, {{1, 6}, {3, 4}, {5, 6}, {6, 7}, {6, 9}}, {{1, 6}, {3, 6}, {4, 5}, {6, 7}, {6, 9}}, {{1, 6}, {3, 6}, {4, 7}, {5, 6}, {6, 9}}, {{1, 6}, {3, 6}, {4, 9}, {5, 6}, {6, 7}}, {{1, 6}, {3, 6}, {5, 6}, {6, 7}, {8, 9}}, {{1, 6}, {3, 6}, {5, 6}, {6, 9}, {7, 8}}, {{1, 6}, {3, 6}, {5, 8}, {6, 7}, {6, 9}}, {{1, 6}, {3, 8}, {5, 6}, {6, 7}, {6, 9}}, {{1, 6}, {3, 8}, {5, 8}, {7, 8}, {8, 9}}, {{1, 8}, {2, 3}, {2, 5}, {2, 7}, {2, 9}}, {{1, 8}, {2, 3}, {5, 8}, {7, 8}, {8, 9}}, {{1, 8}, {2, 5}, {3, 8}, {7, 8}, {8, 9}}, {{1, 8}, {2, 7}, {3, 8}, {5, 8}, {8, 9}}, {{1, 8}, {2, 9}, {3, 8}, {5, 8}, {7, 8}}, {{1, 8}, {3, 4}, {4, 5}, {4, 7}, {4, 9}}, {{1, 8}, {3, 4}, {5, 8}, {7, 8}, {8, 9}}, {{1, 8}, {3, 6}, {5, 6}, {6, 7}, {6, 9}}, {{1, 8}, {3, 6}, {5, 8}, {7, 8}, {8, 9}}, {{1, 8}, {3, 8}, {4, 5}, {7, 8}, {8, 9}}, {{1, 8}, {3, 8}, {4, 7}, {5, 8}, {8, 9}}, {{1, 8}, {3, 8}, {4, 9}, {5, 8}, {7, 8}}, {{1, 8}, {3, 8}, {5, 6}, {7, 8}, {8, 9}}, {{1, 8}, {3, 8}, {5, 8}, {6, 7}, {8, 9}}, {{1, 8}, {3, 8}, {5, 8}, {6, 9}, {7, 8}}]

Now to Split the List L to sublists like this

Now going through L list in details (All my lists will be like this only}

Sublist L1 will have all those from first in this firstly we see  edge {0,1} we take all those which take {0,1} in sequencial manner and put in sublist L1

Now as we proceed we see the next starting edge is {0,3} so their is none with {0,2} so we pick all those with {0,3} in the first sequencial manner and put in sublist L2

Similiar we pick all sublist from this with first element unique in the sequential manner and make a list of lists

Lk:=[L1,L2,L3,L4,....]

Code until L

which was done

Toy_code_(1).mw

Now again proceed we can observe we can see their none with {0,4} next is only with {0,5} 

Then i need write a function F which takes a List say L1 returns

all possible 2 element permutions from L1 list say [S1,S2] and [S2,S1] like that all possible as order also matter where that set is positioned

Given Graph G and Graph H how to store the list of all the various Subgraphs of Shape H in G with their vertex labels.

I have write a code but it may not be the optimal way to do it and it may be time consuming for even medium graphs too

Any way to optimize

Let From this list of subgraphs H of say [G0,G1,G2,.....] we pick NumberOfVertices(G) Graphs
say H= [H0,H1,H2,H3,....,Hn]

i) Now any two graph Hi,Hj have excatly one edge interection if (i,j) is an edge in G

 ii) An edge in  the list H will occur exactly twice only in the entire list of graphs

If these two conditions satifies all such H I need to save

 

Toycode until picking graphs

From that how to pick lists which satify conditions i) and ii) is another need help

Toy_code.mw

Please excuse my thickness if any.

In the following IF statement:

if 1 <> 2 then
    OuterThen;
    if 1 <> 2 then
        InnerThen;
    else
        InnerElse;
    end if;    
else
    OuterElse;
end if;

I expect the output:

OuterThen
InnerThen

but I only get:

OuterThen

Why?

In practice, I need calculate the (principal) squareroot of some suitable large matrix exactly (so the desired result should not involve floating-point numbers, otherwise the decomposition will be of no theoretical value as a certificate …). But I find it difficult to do so in Maple. (What about the Efficient Computations - Maple Help (maplesoft.com)?)
Below are some matrices: (They are not contrived academic examples.) 

M__4 := <1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0|0,3,0,-6,0,3,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,-3,0,0,0,0,0,0,0,0,6,0,0,0,0,-3,0|-2,0,7,0,-8,0,3,0,0,0,0,0,0,0,0,1,0,-7,0,2,0,0,0,0,0,0,4,0,3,0,0,0,0,-3,0,0|0,-6,0,13,0,-8,0,1,0,0,0,0,0,0,0,0,6,0,-4,0,4,0,0,0,0,0,0,-1,0,-11,0,0,0,0,6,0|1,0,-8,0,13,0,-6,0,0,0,0,0,0,0,0,4,0,8,0,-1,0,0,0,0,0,0,-11,0,-6,0,0,0,0,6,0,0|0,3,0,-8,0,7,0,-2,0,0,0,0,0,0,0,0,-3,0,-1,0,1,0,0,0,0,0,0,2,0,4,0,0,0,0,-3,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,0,0,1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,4,0,0,0,-2,0,0,0,0,0,0,0,0,-6,0,6,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,-2,0,-1,0,0,0,0,2,0,2,0,0,-1|0,0,0,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,-6,0,-6,0,0,0,0,0,0,6,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,-2,0,-6,0,4,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,0,0,-3,0,3,0,0,0,0,0,0,6,0,3,0,-6,0,0,0,0,-3,0,3,0,0,0|-2,0,1,0,4,0,-3,0,0,0,0,0,0,0,0,7,0,-1,0,2,0,0,0,0,0,0,-8,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|2,0,-7,0,8,0,-3,0,0,0,0,0,0,0,0,-1,0,7,0,-2,0,0,0,0,0,0,-4,0,-3,0,0,0,0,3,0,0|0,3,0,-4,0,-1,0,2,0,0,0,0,0,0,0,0,-3,0,7,0,-7,0,0,0,0,0,0,-2,0,8,0,0,0,0,-3,0|-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0|0,-3,0,4,0,1,0,-2,0,0,0,0,0,0,0,0,3,0,-7,0,7,0,0,0,0,0,0,2,0,-8,0,0,0,0,3,0|0,0,0,0,0,0,0,0,-6,0,-1,0,-6,0,6,0,0,0,0,0,0,13,0,8,0,-11,0,0,0,0,-8,0,4,0,0,1|0,0,0,0,0,0,0,0,0,-6,0,-6,0,6,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,-2,0,-3,0,3,0,0,0,0,0,0,8,0,7,0,-4,0,0,0,0,-7,0,-1,0,0,2|0,0,0,0,0,0,0,0,0,6,0,-6,0,0,0,0,0,0,0,0,0,0,-6,0,12,0,0,0,0,0,0,-6,0,0,0,0|0,0,0,0,0,0,0,0,6,0,-1,0,6,0,-6,0,0,0,0,0,0,-11,0,-4,0,13,0,0,0,0,4,0,-8,0,0,1|1,0,4,0,-11,0,6,0,0,0,0,0,0,0,0,-8,0,-4,0,-1,0,0,0,0,0,0,13,0,6,0,0,0,0,-6,0,0|0,0,0,-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,6,0,-11,0,4,0,1,0,0,0,0,0,0,0,0,-6,0,8,0,-8,0,0,0,0,0,0,-1,0,13,0,0,0,0,-6,0|0,0,0,0,0,0,0,0,3,0,2,0,3,0,-3,0,0,0,0,0,0,-8,0,-7,0,4,0,0,0,0,7,0,1,0,0,-2|0,0,0,0,0,0,0,0,0,-2,0,6,0,-2,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,4,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,2,0,-3,0,3,0,0,0,0,0,0,4,0,-1,0,-8,0,0,0,0,1,0,7,0,0,-2|0,0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,-6,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,2,0,1,0,0,0,0,-2,0,-2,0,0,1>:
M__3 := 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Unexpectedly, LinearAlgebra[MatrixPower] does not work well. For instance: 
 

restart;

interface(version)

`Standard Worksheet Interface, Maple 2023.0, Windows 10, March 6 2023 Build ID 1689885`

(1)

M__4 := <1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0|0,3,0,-6,0,3,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,-3,0,0,0,0,0,0,0,0,6,0,0,0,0,-3,0|-2,0,7,0,-8,0,3,0,0,0,0,0,0,0,0,1,0,-7,0,2,0,0,0,0,0,0,4,0,3,0,0,0,0,-3,0,0|0,-6,0,13,0,-8,0,1,0,0,0,0,0,0,0,0,6,0,-4,0,4,0,0,0,0,0,0,-1,0,-11,0,0,0,0,6,0|1,0,-8,0,13,0,-6,0,0,0,0,0,0,0,0,4,0,8,0,-1,0,0,0,0,0,0,-11,0,-6,0,0,0,0,6,0,0|0,3,0,-8,0,7,0,-2,0,0,0,0,0,0,0,0,-3,0,-1,0,1,0,0,0,0,0,0,2,0,4,0,0,0,0,-3,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,0,0,1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,4,0,0,0,-2,0,0,0,0,0,0,0,0,-6,0,6,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,-2,0,-1,0,0,0,0,2,0,2,0,0,-1|0,0,0,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,-6,0,-6,0,0,0,0,0,0,6,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,-2,0,-6,0,4,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,0,0,-3,0,3,0,0,0,0,0,0,6,0,3,0,-6,0,0,0,0,-3,0,3,0,0,0|-2,0,1,0,4,0,-3,0,0,0,0,0,0,0,0,7,0,-1,0,2,0,0,0,0,0,0,-8,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|2,0,-7,0,8,0,-3,0,0,0,0,0,0,0,0,-1,0,7,0,-2,0,0,0,0,0,0,-4,0,-3,0,0,0,0,3,0,0|0,3,0,-4,0,-1,0,2,0,0,0,0,0,0,0,0,-3,0,7,0,-7,0,0,0,0,0,0,-2,0,8,0,0,0,0,-3,0|-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0|0,-3,0,4,0,1,0,-2,0,0,0,0,0,0,0,0,3,0,-7,0,7,0,0,0,0,0,0,2,0,-8,0,0,0,0,3,0|0,0,0,0,0,0,0,0,-6,0,-1,0,-6,0,6,0,0,0,0,0,0,13,0,8,0,-11,0,0,0,0,-8,0,4,0,0,1|0,0,0,0,0,0,0,0,0,-6,0,-6,0,6,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,-2,0,-3,0,3,0,0,0,0,0,0,8,0,7,0,-4,0,0,0,0,-7,0,-1,0,0,2|0,0,0,0,0,0,0,0,0,6,0,-6,0,0,0,0,0,0,0,0,0,0,-6,0,12,0,0,0,0,0,0,-6,0,0,0,0|0,0,0,0,0,0,0,0,6,0,-1,0,6,0,-6,0,0,0,0,0,0,-11,0,-4,0,13,0,0,0,0,4,0,-8,0,0,1|1,0,4,0,-11,0,6,0,0,0,0,0,0,0,0,-8,0,-4,0,-1,0,0,0,0,0,0,13,0,6,0,0,0,0,-6,0,0|0,0,0,-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,6,0,-11,0,4,0,1,0,0,0,0,0,0,0,0,-6,0,8,0,-8,0,0,0,0,0,0,-1,0,13,0,0,0,0,-6,0|0,0,0,0,0,0,0,0,3,0,2,0,3,0,-3,0,0,0,0,0,0,-8,0,-7,0,4,0,0,0,0,7,0,1,0,0,-2|0,0,0,0,0,0,0,0,0,-2,0,6,0,-2,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,4,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,2,0,-3,0,3,0,0,0,0,0,0,4,0,-1,0,-8,0,0,0,0,1,0,7,0,0,-2|0,0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,-6,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,2,0,1,0,0,0,0,-2,0,-2,0,0,1>:

M__3 := 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timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(M__4, 1/2)))

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 10 x 10 matrix

 

IntegerCharacteristicPolynomial: Used total of  2  prime(s)

 

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 10 x 10 matrix

 

IntegerCharacteristicPolynomial: Used total of  2  prime(s)

 

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 10 x 10 matrix

 

IntegerCharacteristicPolynomial: Used total of  2  prime(s)

 

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 6 x 6 matrix

 

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

 

Error, (in simplify/sqrt/fraction) time expired

 

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(M__3, 1/2)))

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 13 x 13 matrix

 

IntegerCharacteristicPolynomial: Used total of  4  prime(s)

 

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

 

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

 

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

 

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

 

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

 

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

 

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

 

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

 

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

 

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

 

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

 

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

 

memory used=2.84GiB, alloc change=8.00MiB, cpu time=3.05m, real time=3.01m, gc time=5.84s

 

_rtable[36893490642867319380]

(2)

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(M__2, 1/2)))

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 6 x 6 matrix

 

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

 

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 6 x 6 matrix

 

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

 

memory used=2.40GiB, alloc change=0 bytes, cpu time=2.85m, real time=2.82m, gc time=4.08s

 

_rtable[36893490642428505372]

(3)

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(M__1, 1/2)))

CharacteristicPolynomial: working on determinant of minor 2

 

CharacteristicPolynomial: working on determinant of minor 3

 

CharacteristicPolynomial: working on determinant of minor 4

 

CharacteristicPolynomial: working on determinant of minor 5

 

CharacteristicPolynomial: working on determinant of minor 6

 

CharacteristicPolynomial: working on determinant of minor 2

 

CharacteristicPolynomial: working on determinant of minor 3

 

CharacteristicPolynomial: working on determinant of minor 4

 

CharacteristicPolynomial: working on determinant of minor 5

 

CharacteristicPolynomial: working on determinant of minor 6

 

CharacteristicPolynomial: working on determinant of minor 2

 

CharacteristicPolynomial: working on determinant of minor 3

 

CharacteristicPolynomial: working on determinant of minor 4

 

CharacteristicPolynomial: working on determinant of minor 5

 

CharacteristicPolynomial: working on determinant of minor 6

 

CharacteristicPolynomial: working on determinant of minor 2

 

CharacteristicPolynomial: working on determinant of minor 3

 

CharacteristicPolynomial: working on determinant of minor 4

 

CharacteristicPolynomial: working on determinant of minor 5

 

CharacteristicPolynomial: working on determinant of minor 6

 

CharacteristicPolynomial: working on determinant of minor 2

 

CharacteristicPolynomial: working on determinant of minor 3

 

CharacteristicPolynomial: working on determinant of minor 4

 

CharacteristicPolynomial: working on determinant of minor 5

 

CharacteristicPolynomial: working on determinant of minor 6

 

Error, (in LinearAlgebra:-MatrixFunction) time expired

 

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalhf[hfloat](M__4), 1/2)))

Eigenvalues: calling external function

 

Eigenvalues: CLAPACK hw_dgeevx_

 

memory used=468.66MiB, alloc change=0 bytes, cpu time=18.48s, real time=19.71s, gc time=1.09s

 

_rtable[36893490642747474396]

(4)

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalhf[hfloat](M__3), 1/2)))

Eigenvalues: calling external function

 

Eigenvalues: CLAPACK hw_dgeevx_

 

Multiply: copying first Matrix to enable external call

 

Multiply: calling external function

 

Multiply: NAG hw_f06yaf

 

unknown: NAG hw_f06yaf

 

unknown: NAG hw_f06yaf

 

unknown: NAG hw_f06yaf

 

memory used=13.43GiB, alloc change=0 bytes, cpu time=8.66m, real time=8.46m, gc time=26.41s

 

_rtable[36893490642773280876]

(5)

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalhf[hfloat](M__2), 1/2)))

Eigenvalues: calling external function

 

Eigenvalues: CLAPACK hw_dgeevx_

 

Multiply: copying first Matrix to enable external call

 

Multiply: calling external function

 

Multiply: NAG hw_f06yaf

 

unknown: NAG hw_f06yaf

 

memory used=13.45GiB, alloc change=0 bytes, cpu time=8.39m, real time=8.23m, gc time=26.12s

 

_rtable[36893490642902286332]

(6)

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalhf[hfloat](M__1), 1/2)))

Eigenvalues: calling external function

 

Eigenvalues: CLAPACK hw_dgeevx_

 

Multiply: copying first Matrix to enable external call

 

Multiply: calling external function

 

Multiply: NAG hw_f06yaf

 

unknown: NAG hw_f06yaf

 

Error, (in LinearAlgebra:-MatrixFunction) time expired

 

"Digits+=5:"timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalf(M__4), 1/2)))

Eigenvalues: calling external function

 

Eigenvalues: copying first Matrix, to enable external call

 

Eigenvalues: CLAPACK hw_dgeevx_

 

memory used=418.17MiB, alloc change=0 bytes, cpu time=16.33s, real time=16.16s, gc time=796.88ms

 

_rtable[36893490642540182332]

(7)

"Digits+=5:"timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalf(M__1), 1/2)))

Eigenvalues: calling external function

 

Eigenvalues: copying first Matrix, to enable external call

 

Eigenvalues: CLAPACK sw_dgeevx_

 

Multiply: copying first Matrix to enable external call

 

Multiply: copying second Matrix to enable external call

 

Multiply: calling external function

 

Multiply: NAG sw_f06yaf

 

unknown: copying second Matrix to enable external call

 

unknown: calling external function

 

unknown: NAG sw_f06yaf

 

Error, (in LinearAlgebra:-MatrixFunction) time expired

 

time() = 5731.593NULL


 

Download performance_of_`sqrtm`.mw

The last six numerical experiments all fail to compute the square root. (Accordingly, here it is impossible to convert each numeric element to one of the "simplest" algebraic numbers that approximates it well.) As you can see, if I execute them directly (i.e., without converting each entry to the nearest floating-point value), the elapsed time required to run the procedure is still unacceptable! (Note that they can be evaluated symbolically in fact.) Is this a bug of Maple? And how do I get the desired results in Maple efficiently?

Hi,

I want to illustrate the intersection between The ellipsoid   "x^2/4 + y^2/9 + z^2/16 = 1"and the plane  "z = 3"

Normally, I should obtain an ellipse G1, but its graph does not match the graph obtained ? Thanks for your ideas
 

G2Exercice1.mw

How to solve this RLC Electric Circuit  with this initial conditions,I couldn't plot this equation.Help me to do this problem.

EC-2.mw

Let y>1. It can be proved (maybe by hand) that the following four expressions are mathematically equivalent: 

assume(y > 1); # Assumption!
expr := [0, 0, 0, 0]: # Preallocation.
expr[1] := exp(y*LambertW(ln(y))):
expr[2] := (ln(y)/LambertW(ln(y)))^y:
expr[3] := eval(x^(x^x), x = exp(LambertW(ln(y)))):
expr[4] := eval(x^(x^x), x = ln(y)/LambertW(ln(y))):

But unfortunately, when I tried to simplify expri - exprj (symbolically), I just got: 

seq(seq(ifelse(j <> i, [i, j, verify(expr[j], expr[i], equal)], NULL), j = 1 .. numelems(expr)), i = 1 .. numelems(expr)); # is(expr[j] = expr[i]) does not work as well.
 = 
   [1, 2, FAIL], [1, 3, FAIL], [1, 4, FAIL], [2, 1, FAIL], 

     [2, 3, FAIL], [2, 4, FAIL], [3, 1, FAIL], [3, 2, FAIL], 

     [3, 4, true], [4, 1, FAIL], [4, 2, FAIL], [4, 3, true]


In other words, Maple can only determine that expr[3] = expr[4].
One may check that, for example, 

MmaTranslator:-Mma:-Chop([seq](seq(evalhf(subs(y = log10(rand()), expr[i] - expr[j])), j = 1 .. numelems(expr)), i = 1 .. numelems(expr)), 2^(-26));
 = 
        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

However, the desired approach is simplifying them symbolically. Is there a way to do so in Maple?

 

Hi, I'm trying to add the display of graphs G1, G2, and G3 (representing the intersections -AnimCompl file-) to my three animations (G5Ex2-file). I've tried using the background option and the display command, but without success. Thank you for your help.

AnimCompl.mw

G5Ex2.mw

I want to put one option of tetrahedron SABC with AB=c, BC=a, AC=b, SA=m, SB=n, SC=p. I tried
restart;
with(Student:-MultivariateCalculus);
A := [0, 0, 0];
B := [c, 0, 0];
C := [(-a^2 + b^2 + c^2)/(2*c), sqrt((a + b - c)*(a - b + c)*(-a + b + c)*(a + b + c))/(2*c), 0];
S := [x, y, z];
solve([distance(S, A) = m, distance(S, B) = n, distance(S, C) = p], [x, y, z]);

I got warning, solutions may have been lost. How can I solve that system of equations?

Just need some help using URL:-Get

Get("https://sdo.gsfc.nasa.gov/assets/img/browse/2023/05/05/20230505 _184918_512_0304.jpg")

I'm sure there are some tags to use but not sure how.  The site does show a script on best practices, but at the moment don't know how to apply them.  Can anyone offer some help?

Hi every one, I am trying to find a relation from an equation by putting it is equal to zero but could not get the solution.

Solution_of_an_Equation.mw

i want to plot form 15 to 0 and 5 to 1.....but my plot shows only increasing order..how can i change it?

Download plot.mw

Is there a way to suppress dependencies of a all functions appearing in a worksheet without having to do suppress([each function])?

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