Maple 2023 Questions and Posts

These are Posts and Questions associated with the product, Maple 2023

I would like to collect in derivatives of a function(s) in a worksheet where I have the Physics package loaded but It does not want to do it. When I bring the expression to a blank worksheet it does everything just fine. Am I missing something small? Or is there additional commands/options I need to input? 

Thoughts?

CollectDiff.mw

The highly oscillatory integrand is: 

(sin(x + sqrt(x)) + x*BesselJ(0, x^2))/(1 + x): # int(%, x = 0 .. infinity, numeric); # Note that the upper limit of `x` is NOT finite.

Unfortunately, Maple still cannot return a result in a few minutes. For instance, 

restart;
infolevel[`evalf/int`] := 1:

smartplot((sin(x+sqrt(x))+x*BesselJ(0, x^2))/(1+x))

 

timelimit(0.1e3, evalf(Int((sin(x+sqrt(x))+x*BesselJ(0, x^2))/(1+x), x = 0 .. infinity)))

Control: Entering NAGInt

trying d01amc (nag_1d_quad_inf)
Control: d01amc failed
evalf/int/control: NAG failed result = result
evalf/int/improper: integrating on interval 0 .. infinity
evalf/int/improper: applying transformation x = 1/x
evalf/int/improper: interval is 0 .. 1 for the integrand:

                        /     (1/2)\            /    2\
                     sin\x + x     / + x BesselJ\0, x /
                     ----------------------------------
                                   1 + x               

and interval is 0 .. 1 for the integrand:

                                                /   1 \
                                         BesselJ|0, --|
                        /       (1/2)\          |    2|
                        |1   /1\     |          \   x /
                     sin|- + |-|     | + --------------
                        \x   \x/     /         x       
                     ----------------------------------
                                 /    1\  2            
                                 |1 + -| x             
                                 \    x/               

Control: Entering NAGInt
Control: trying d01ajc (nag_1d_quad_gen)
Control: d01ajc failed
evalf/int/control: NAG failed result = result

evalf/int/control: singularity at left end-point
evalf/int/transform: series contains {x^(1/2), x^(3/2), x^(5/2), x^(7/2), x^(9/2), x^(11/2), x^(13/2), x^(15/2), x^(17/2), x^(19/2), x^(21/2), x^(23/2), x^(25/2)}

evalf/int/singleft: applying transformation x = x^2
evalf/int/singleft: interval is 0 .. 1. for the integrand:

                  /   / 2            \    2        /    4\\  
                2 \sin\x  + csgn(x) x/ + x  BesselJ\0, x // x
                ---------------------------------------------
                                        2                    
                                   1 + x                     

evalf/int/control: Applying simplify/ln, integrand is 2*(sin(x^2+x)+x^2*BesselJ(0,x^4))/(1+x^2)*x
evalf/int/CreateProc: Trying easyproc
evalf/int/CreateProc: Trying makeproc
evalf/int/ccquad: n = 2 integral estimate = .7758263476953
                    n = 6 integral estimate = .8097413335347
evalf/int/ccquad: n = 18 integral estimate = .8097470386638
                                  error = .8097470386638e-11
From ccquad, result = .8097470386638 integrand evals = 19 error = .8097470386638e-11
Control: Entering NAGInt
Control: trying d01ajc (nag_1d_quad_gen)
Control: d01ajc failed
evalf/int/control: NAG failed result = result

evalf/int/control: singularity at left end-point

evalf/int/transform: series contains {1/x^(3/2), x^(1/2), cos(1/x^2+1/4*Pi), sin((x+x^(1/2))/x^(3/2)), sin(1/4*(4+Pi*x^2)/x^2)}
evalf/int/singleft: applying transformation x = x^2
evalf/int/singleft: interval is 0 .. 1. for the integrand:

                   /   /        /1\  \                    \
                   |   |1 + csgn|-| x|                    |
                   |   |        \x/  |  2          /   1 \|
                 2 |sin|-------------| x  + BesselJ|0, --||
                   |   |      2      |             |    4||
                   \   \     x       /             \   x //
                 ------------------------------------------
                                 3 /     2\                
                                x  \1 + x /                

evalf/int/control: Applying simplify/ln, integrand is 2*(sin((1+x)/x^2)*x^2+BesselJ(0,1/x^4))/x^3/(1+x^2)
evalf/int/control: Applying simplify/trig, integrand is (2*sin((1+x)/x^2)*x^2+2*BesselJ(0,1/x^4))/(x^3+x^5)
evalf/int/control: singularity at left end-point

evalf/int/transform: series contains {cos(1/x^4+1/4*Pi), sin((1+x)/x^2), sin(1/4*(4+Pi*x^4)/x^4)}
evalf/int/transform: no transform found
evalf/int/series: integrating on 0 .. .2493468135214 the series:

       2 (%2)     2 (%2) x   /2 (%2)      \  3   /  2 (%2)      \  5   
      --------- - -------- + |------- - %3| x  + |- ------- + %3| x    
        (1/2)       (1/2)    |  (1/2)     |      |    (1/2)     |      
      Pi      x   Pi         \Pi          /      \  Pi          /      
                                                                       
           /       2 (%2)      \  7   /     2 (%2)      \  9           
         + |- %3 + ------- - %4| x  + |%3 - ------- + %4| x            
           |         (1/2)     |      |       (1/2)     |              
           \       Pi          /      \     Pi          /              
                                                                       
                                                                       
           /2 (%2)           \  11   /          2 (%2) \  13           
         + |------- - %4 - %5| x   + |%4 + %5 - -------| x             
           |  (1/2)          |       |            (1/2)|               
           \Pi               /       \          Pi     /               
                                                                       
           /       2 (%2)      \  15   /     2 (%2)      \  17    / 19\
         + |- %5 + ------- + %6| x   + |%5 - ------- - %6| x   + O\x  /
           |         (1/2)     |       |       (1/2)     |             
           \       Pi          /       \     Pi          /             
                                                                       
                                                                       
                      /        4\                                      
             (1/2)    |4 + Pi x |                                      
      %1 := 2      sin|---------|                                      
                      |     4   |                                      
                      \  4 x    /                                      
                    /1 + x\   (1/2)                                    
      %2 := %1 + sin|-----| Pi                                         
                    |  2  |                                            
                    \ x   /                                            
             (1/2)    /1    1   \                                      
            2      cos|-- + - Pi|                                      
                      | 4   4   |                                      
                      \x        /                                      
      %3 := ---------------------                                      
                      (1/2)                                            
                  4 Pi                                                 
                        /        4\                                    
               (1/2)    |4 + Pi x |                                    
            9 2      sin|---------|                                    
                        |     4   |                                    
                        \  4 x    /                                    
      %4 := -----------------------                                    
                       (1/2)                                           
                  64 Pi                                                
                (1/2)    /1    1   \                                   
            53 2      cos|-- + - Pi|                                   
                         | 4   4   |                                   
                         \x        /                                   
      %5 := ------------------------                                   
                        (1/2)                                          
                  512 Pi                                               
                           /        4\                                 
                  (1/2)    |4 + Pi x |                                 
            1371 2      sin|---------|                                 
                           |     4   |                                 
                           \  4 x    /                                 
      %6 := --------------------------                                 
                          (1/2)                                        
                  16384 Pi                                             

evalf/int/CreateProc: Trying easyproc
evalf/int/CreateProc: Trying makeproc

evalf/int/CreateProc: Trying procmake
evalf/int/control: applying double-exponential method
evalhf mode unsuccessful -- retry in software floats
evalf/int/quadexp: applying double-exponential method

Error, (in tools/sign) time expired

 

time() =

3999.500

(1)

 =

%?

Download oscillatoryInt.mws

I tried to specify another method (as described in evalf/int), but it seems that this doesn't work. Any ideas?

Hello,

How do I isolate the variable ng in the following expression:

rho := `ρg`*ng + `ρp`*np + `ρw`*nw;
B := np/Kp + nw/Kw + ng/Kg;

C := (B/rho)^(1/2);

As you can see, I am running the isolate command, but it is not including the variable C, or it returns the wrong result.

isolate(((np/Kp + nw/Kw + ng/Kg)/(`ρg`*ng + `ρp`*np + `ρw`*nw))^(1/2), ng);
                               ( Kp nw + Kw np) Kg
                   ng = -     -----------------

                                          Kw Kp       

 

isolate(C=((np/Kp + nw/Kw + ng/Kg)/(`ρg`*ng + `ρp`*np + `ρw`*nw))^(1/2), ng);
                             0 = 0

Thanks.                

In certain tasks, I need to find all accurate positive (not just nonnegative) roots that exist of some multivariate polynomials like: 

nsd := 16*a*b*c*(9 + a^2 + b^2 + c^2)*(b*c + a*(b + c) + 3*(a + b + c)) - (3 + a + b + c)^2*(a*b + 3*c)*(3*b + a*c)*(3*a + b*c): # assume((a, b, c) >~ 0);

According to fsolve/details, for one general equation, the fsolve command only computes "a single real root", so it is inadequate to tackle this question. But if I use the solve command with floating-point values, the computation cannot finish in ten minutes instead! (Maybe a longer time will suffice, yet this is rather unacceptable.) 

restart;

"Digits+=Digits:"

#assume((a, b, c) >~ 0);
nsd := 16*a*b*c*(a^2 + b^2 + c^2 + 9)*(a*(b + c) + 3*(a + b + c) + b*c) - (a + b + c + 3)^2*(a*b + 3*c)*(a*c + 3*b)*(b*c + 3*a):

fsolve({`~`[`>`](a, b, c, ` $`, 0), nsd = 0}, fulldigits)

Error, (in fsolve) expecting an equation or set or list of equations, but received inequalities {16*a*b*c*(a^2+b^2+c^2+9)*(a*(b+c)+3*a+3*b+3*c+b*c)-(a+b+c+3)^2*(a*b+3*c)*(a*c+3*b)*(b*c+3*a) = 0, 0 < a, 0 < b, 0 < c}

 

fsolve([`$`(nsd = 0, 3)], fulldigits, {`~`[`=`](a, b, c, ` $`, 0 .. infinity)}, avoid = {{a = 0}, {b = 0}, {c = 0}})eval(nsd, %)

{a = 3.0000000005817604971, b = 3.0000000004676996798, c = 3.0000000004610312655}

(1)

timelimit(0.6e3, RealDomain[solve]([`~`[`>`](a, b, c, ` $`, 0), nsd = 0.], allsolutions))

Error, (in gcd/gcdchrem1) time expired

 

timelimit(0.6e3, `assuming`([solve(nsd = 0., useassumptions, allsolutions)], [`~`[`>`](a, b, c, ` $`, 0)]))

Error, (in modp1/DistDeg) time expired

 

""(* However, there are (at least) five positive solutions to 'nsd = 0'. *)" map2(eval,nsd,[{a=1,b=1,c=1},{a=3,b=3,c=3},{a=3,b=3,c=9},{a=3,b=9,c=3},{a=9,b=3,c=3}])"

[0, 0, 0, 0, 0]

(2)

time()

11127.031

(3)

NULL


Download solve_numerically.mws

Is there any workaround to obtain those (finitely many) positive solutions completely?

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?