preview is not very useful usually - it shows a different output - it looks as if it uses different css sheets. 

Such things didn't happen to me often - and I don't usually bother to change the input format - it is usually OK. But once, I remember, it was changed exactly when I used the preview. So maybe the previewing of the post changes its input format.

Alec

preview is not very useful usually - it shows a different output - it looks as if it uses different css sheets. 

Such things didn't happen to me often - and I don't usually bother to change the input format - it is usually OK. But once, I remember, it was changed exactly when I used the preview. So maybe the previewing of the post changes its input format.

Alec

I changed my username there to Alec Mihailovs, and it crashed right after that - perhaps, that may be the cause - changing it in so many places..., and maybe, a space in it.

Alec

That's an old problem of a computer representation of numbers - if they are represented in a binary form, that differs from their decimal representation. For example,

evalhf(0.10);

                         0.100000000000000004

To avoid problems related to that, one can increase Digits,

Digits:=16;

                             Digits := 16

A:=Matrix([[-0.30,0.20,0.10],[0.20,-0.30,0.10],[0.10,0.10,-0.20]]);

                         [-0.30    0.20     0.10 ]
                         [                       ]
                    A := [0.20     -0.30    0.10 ]
                         [                       ]
                         [0.10     0.10     -0.20]

LinearAlgebra:-ReducedRowEchelonForm(A);

                  [1.    -0.    -1.000000000000000]
                  [                               ]
                  [0.    1.     -1.000000000000000]
                  [                               ]
                  [0.    0.             0.        ]

Another way of dealing with that is setting UseHardwareFloats to false,

restart;
A:=Matrix([[-0.30,0.20,0.10],[0.20,-0.30,0.10],[0.10,0.10,-0.20]]):
UseHardwareFloats:=false:
LinearAlgebra:-ReducedRowEchelonForm(A);

                     [1.    -0.    -1.000000000]
                     [                         ]
                     [0.    1.     -1.000000000]
                     [                         ]
                     [0.    0.          0.     ]

Alec

Similarly - keep track of the sign and print if it changes,

x:=1: sol:=solve(y=x-3): s:=signum(sol):
for x from 2 to 10 do 
sol1:=solve(y=x-3); s1:=signum(sol1); 
if s1<>s then 
if x-1<>printed then print(x-1,sol) fi;
printed:=x; print(x,sol1) fi;
sol:=sol1; s:=s1 od:

                                2, -1
                                 3, 0
                                 4, 1

Alec

Similarly - keep track of the sign and print if it changes,

x:=1: sol:=solve(y=x-3): s:=signum(sol):
for x from 2 to 10 do 
sol1:=solve(y=x-3); s1:=signum(sol1); 
if s1<>s then 
if x-1<>printed then print(x-1,sol) fi;
printed:=x; print(x,sol1) fi;
sol:=sol1; s:=s1 od:

                                2, -1
                                 3, 0
                                 4, 1

Alec

plots look the same (if move Maple's plot up by 1).

(1,0) is the derivative over the first variable. Hypergeometric0F1Regularized(b,x) is hypergeom([],[b],x)/GAMMA(b) in Maple if b is not a nonpositive integer, and the limit of that if it is.

Alec

Replacing evalf with evalhf speeds things approximately 4 times. There are some cases where evalf can be used and evalhf - can't though.

The following also works fast,

x:=[Pi,sqrt(2), sqrt(3), sqrt(5), sqrt(10), sqrt(7)]:
sort([seq([evalhf(i),i],i=x)])[..,2];

                   1/2   1/2   1/2   1/2        1/2
                 [2   , 3   , 5   , 7   , Pi, 10   ]

time(seq(sort([seq([evalhf(i),i],i=x)])[..,2],j=1..100000));

                                4.414

time(seq(attributes~(sort([seq(setattribute(evalhf(i),i),i=x)])),
j=1..100000));
                                4.976

Alec

Replacing evalf with evalhf speeds things approximately 4 times. There are some cases where evalf can be used and evalhf - can't though.

The following also works fast,

x:=[Pi,sqrt(2), sqrt(3), sqrt(5), sqrt(10), sqrt(7)]:
sort([seq([evalhf(i),i],i=x)])[..,2];

                   1/2   1/2   1/2   1/2        1/2
                 [2   , 3   , 5   , 7   , Pi, 10   ]

time(seq(sort([seq([evalhf(i),i],i=x)])[..,2],j=1..100000));

                                4.414

time(seq(attributes~(sort([seq(setattribute(evalhf(i),i),i=x)])),
j=1..100000));
                                4.976

Alec

Sorting with attributes, invented by Joe Riel, was a brilliant idea!

Very useful in many cases.

Alec

Sorting with attributes, invented by Joe Riel, was a brilliant idea!

Very useful in many cases.

Alec

Halley's method also could be used - it's convergence rate is cubic (Newton's method - quadratic).

But - both methods use evaluations of exponents - so the asympt (also using exponents) seems to be more simple to use.

Frankly, I don't understand what is wrong with using LambertW "on paper" - the next step would be replasing all sines, cosines, square roots, logarithms and exponents with some partial sums of their series, too?

Alec

Halley's method also could be used - it's convergence rate is cubic (Newton's method - quadratic).

But - both methods use evaluations of exponents - so the asympt (also using exponents) seems to be more simple to use.

Frankly, I don't understand what is wrong with using LambertW "on paper" - the next step would be replasing all sines, cosines, square roots, logarithms and exponents with some partial sums of their series, too?

Alec

Mathematica (but not Wolfram Alpha) gives

In[1]:= Integrate[BesselK[1, t]*t^2, {t, 0, x}, 
    Assumptions -> x > 0]

Out[1]= 2 + 2 BesselI[0, x] - 2 x BesselI[1, x] + 
    x^2 BesselI[2, x] Log[x/2] + 
    4 Hypergeometric0F1Regularized(1,0)[-1, x^2/4]

Alec

In Mathematica, built-in functions start with a capital letter and function arguments or parameters are included in square brackets, so sin(x) should be Sin[x], and, say, cos(1) should be Cos[1].

Alec