Author: Frank Postel Date : 3 May 1996 Format: dvi, postscript
In this paper it is demonstrated how MuPAD 1.3 solves 31 problems of the testsuite and which problems MuPAD 1.3 can't solve. Please note that MuPAD is the only non-commercial system of the compared one. It can be used by research institutions and training centres at no change.
>> MS:= SquareMatrix( 30,Rational ): H := MS( func(1/(i+j-1),i,j) ): >> iH:= H^(-1): iH[30,30]; 53338390370510951379337308012614400
+- -+ | a, 1, 0 | | 1, 1, 1 | | 1, 1, 2 | +- -+ >> MExpr:= Matrix( ExpressionField(normal) ): A:= MExpr( [[a,1,0],[1,1,1],[1,1,2]] ); +- -+ | a, 1, 0 | | | | 1, 1, 1 | | | | 1, 1, 2 | +- -+ >> le:= linalg::eigenValues( A ):The following outputs were cutted due to the long results:
>> op( le, 1 ); / / | 2 1/2 1/2 2 1/2 | | 3 a - a + 9 I 3 + (- 3 I) a 3 + I a 3 + 18 | ... \ \ \ / / 2 3 | | | a a ... - 9 | / | 18 | - -- + -- + / \ \ 6 27 1/2 2 3 4 1/2 \1/3 \ 3 (108 a - 90 a + 32 a - 5 a - 81) | | ------------------------------------------- + 1/2 | | 18 / / >> op( le, 2 ); / | 2 1/2 1/2 2 1/2 | 3 a - a + (- 9 I) 3 + 3 I a 3 + (- I) a 3 + ... \ \ / / 2 3 | | | a a ... - 9 | / | 18 | - -- + -- + / \ \ 6 27 1/2 2 3 4 1/2 \1/3 \ 3 (108 a - 90 a + 32 a - 5 a - 81) | | ------------------------------------------- + 1/2 | | 18 / / >> op( le, 3 ); / | 2 1/2 1/2 2 1/2 | 3 a - a + (9 I) 3 + (- 3 I) a 3 + (I) a 3 + ... \ 1/2 2 3 4 1/2 \ 1/3 \ 3 (108 a - 90 a + 32 a - 5 a - 81) | | ... + ------------------------------------------- + 1/2 | | 18 / /
>> radsimp( sqrt( 9+4*sqrt(5) ) ); 1/2 5 + 2
>> e:= (sin(2*x) - sin(x)*cos(x))/(cos(x)*(1+tan(x)^2)): combine( normal(expand(e)), sincos ); sin(x) sin(3 x) ------ + -------- 4 4
>> simplify( (exp(x) - 1)/(exp(x/2)+1),exp ); / x \ exp| - | - 1 \ 2 /
>> solve( x^4 + a*x^2 + 1 = 0, x); { 1/2 2 1/2 1/2 1/2 2 1/2 1/2 { 2 (- a - (a - 4) ) 2 (- a - (a - 4) ) { - ---------------------------, ---------------------------, { 2 2 1/2 2 1/2 1/2 1/2 2 1/2 1/2 } 2 (- a + (a - 4) ) 2 (- a + (a - 4) ) } - ---------------------------, --------------------------- } 2 2 }
>> solve( 2*sin(x) + 5*cos(x) = 3, x ); { / / 1/2 \ / 1/2 \ { | | 5 | | 5 | { | 2 atan| ---- + 1/4 |, 2 PI + 2 atan| ---- + 1/4 |, \ \ 4 / \ 4 / / 1/2 \ \ / | 5 | | | - 2 PI + 2 atan| ---- + 1/4 |, ... |, | \ 4 / / \ / 1/2 \ / 1/2 \ | 5 | | 5 | 2 atan| - ---- + 1/4 |, 2 PI + 2 atan| - ---- + 1/4 |, \ 4 / \ 4 / / 1/2 \ \ } | 5 | | } - 2 PI + 2 atan| - ---- + 1/4 |, ... | } \ 4 / / }Please note that in version 1.3, so-called {\it discrete sets} are used to represent infinite sets of solutions:
>> domtype( op(S,1) ); Dom::DiscreteSet
>> solve( exp(2*x) + exp(x) - 1 = 0, x ); { / / 1/2 \ / 1/2 \ { | | 5 | | 5 | { | ln| ---- - 1/2 |, (- 2 I) PI + ln| ---- - 1/2 |, { \ \ 2 / \ 2 / / 1/2 \ \ / / 1/2 \ | 5 | | | | 5 | 2 I PI + ln| ---- - 1/2 |, ... |, | ln| - ---- - 1/2 |, \ 2 / / \ \ 2 / / 1/2 \ | 5 | (- 2 I) PI + ln| - ---- - 1/2 |, \ 2 / / 1/2 \ \ } | 5 | | } 2 I PI + ln| - ---- - 1/2 |, ... | } \ 2 / / }
x*y^2 + 2*c + x*z = 1 x*y - x^2 + 2*z = 0 x*y - z = 0
>> solve( {x*y^2+2*c+x*z=1, x*y-x^2+2*z=0, x*y-z=0}, {x,y,z} ); { -- 2 -- } { | x x 3 | } { | y = -, z = --, x = RootOf(18 c + 4 x - 9, x) | } { -- 3 3 -- }
>> s:= series( cosh(x)/sinh(x)-1/x,x,5 ); x 3 - + O(x ) 3 >> testtype( s,Type::Series( Taylor ) ); TRUE
Unable to do in MuPAD 1.3
>> sum( 1/(k^2+a^2), k=1..infinity ); / | 2 1/2 2 1/2 2 1/2 | psi((- a ) + 1) - psi(- (- a ) + 1) + 2 (- a ) | \ / 2 1/2 2 1/2 \ \ | - psi(k + (- a ) ) + psi(k - (- a ) ) | | limit| -----------------------------------------, k = infinity | | / | 2 1/2 | | \ 2 (- a ) / / 2 1/2 (2 (- a ) )
>> limit( asinh((u+a)/b) - asinh((u-a)/b), u=infinity ); 0
>> limit( exp(1/x^2)/(exp(1/x^2)+exp(1/x^4)) , x=0 ); 0
>> diff( (-1/5*x^2 + 1/5*x^2*tan(1/2*ln(x))^2 + 4/5*x^2*tan(1/2*ln(x))) / (1 + tan(1/2*ln(x))^2) , x ); / / ln(x) \ / ln(x) \2 | 8 x tan| ----- | 2 x tan| ----- | | 2 x \ 2 / \ 2 / | - --- + ---------------- + ----------------- + ... | 5 5 5 | \ / 2 / ln(x) \ 2 / ln(x) \2 \ \ | 2 4 x tan| ----- | x tan| ----- | | | | x \ 2 / \ 2 / | | ... | - -- + ----------------- + ---------------- | | \ 5 5 5 / / / / ln(x) \2 / / ln(x) \2 \2 \ / | x cos| ----- | | tan| ----- | + 1 | | \ \ 2 / \ \ 2 / / /The output was cutted due to the long result.
Unable to do in MuPAD 1.3
>> int( (2*x^4 + 1)/((x^5+x)*sqrt(x^4+1)) , x ); / 4 \ | 2 x + 1 | int| --------------------, x | | 5 4 1/2 | \ (x + x ) (x + 1) /As one can see, by now the integrator of MuPAD is not able to handle certain algebraic dependences of functions, but we hope that this will be implemented in a next MuPAD release.
>> int( exp(-x)*sin(x)*cos(x) , x ); cos(2 x) sin(2 x) - -------- - --------- 5 exp(x) 10 exp(x)
>> int( x^3/(exp(x)+1), x=0..infinity ); 4 7 PI ----- 120
>> int( ln(x)^2/sqrt(1-x^2), x=0..1 ); 3 2 PI PI ln(2) --- + --------- 24 2
>> laplace( sin(t), t, s ); 1 ------ 2 s + 1
>> ilaplace( %, s, t ); sin(t)
>> laplace( heaviside(t-2) * sin(t), t, s ); / cos(2) s sin(2) \ exp(- 2 s) | ------ + -------- | | 2 2 | \ s + 1 s + 1 /
>> ilaplace( %, s, t ); heaviside(t - 2) (cos(2) sin(t - 2) + sin(2) cos(t - 2))Use combine to get back the original function:
>> combine(%,sincos); sin(t) heaviside(t - 2)
>> fourier( exp(-t^2),t,x ); / 2 \ 1/2 | x | PI exp| - -- | \ 4 /
>> ifourier( exp(-t^2),t,x ); / 2 \ | x | exp| - -- | \ 4 / ----------- 1/2 2 PI
>> solve( ode( x^2*diff(y(x),x)-x*y(x) = x*ln(x) , y(x) ) ); {C1 x - ln(x) - 1}
>> solve( ode(diff(y(x),x) - y(x)^2 - 3*y(x) + 4 = 0, y(x)) ); / ln(y - 1) ln(y + 4) \ solve| --------- - --------- = C2 + x, y, 1, PrincipalValue | \ 5 5 /We consider this example as not solved by MuPAD 1.3.
>> solve( ode( diff(y(x),x$2) - 4*y(x) = sin(x) , y(x)) ); { sin(x) } { - ------ + C3 exp(2 x) + C4 exp(- 2 x) } { 5 }
>> solve( ode( diff(y(x),x$2) - 2/x*diff(y(x),x) + 3/x^2*y(x) = 2*x - 1, y(x) ) ); -- 3 / 1/2 \ / 1/2 \ -- | 2 2 x 3/2 | ln(x) 3 | 3/2 | ln(x) 3 | | | - x + ---- + C2 x cos| ---------- | + C3 x sin| ---------- | | -- 3 \ 2 / \ 2 / --
>> solve( ode( 8*diff(y(x),x$2) + 9*diff(y(x),x)^4 = 0, y(x) ) ); { 3 / 1/2 \ / 1/2 \ } { 2 2 x 3/2 | ln(x) 3 | 3/2 | ln(x) 3 | } { - x + ---- + x C5 cos| ---------- | - x C6 sin| ---------- | } { 3 \ 2 / \ 2 / }
>> solve( ode( 4*diff(y(x),x$4) - 12*diff(y(x),x$3) + 11*diff(y(x),x$2) - 3*diff(y(x),x) = 4*cos(x) , y(x) ) ); / / 1 \ | | -------------------------------------, y | {C14} union solve| int| 2 2 | = \ \ RootOf(- 9 y u1 + 8 u1 C13 + 4, u1) / \ | x + C15, y, 1, PrincipalValue | /We consider this example as not solved by MuPAD 1.3.
x' - x + y' + 2 y = 1 + e^t y' + 2 y + z' + z = 2 + e^t x' - x + z' + z = 3 + e^t
>> solve( ode( { diff(x(t),t) - x(t) + diff(y(t),t) + 2*y(t) = 1+exp(t), diff(y(t),t) + 2*y(t) + diff(z(t),t) + z(t) = 2+exp(t), diff(x(t),t) - x(t) + diff(z(t),t) + z(t) = 3+exp(t) }, {x(t), y(t), z(t)} ) ); { { exp(t) t exp(t) { { z(t) = ------ + C21 exp(-t) + 2, x(t) = -------- + C20 exp(t) - 1, { { 4 2 exp(t) } } y(t) = ------ + C22 exp(- 2 t) } } 6 } }
>> export( plotlib ):
Unable to do in MuPAD 1.3In the current version of MuPAD, the plot command can not handle singularities, therefore a plot of this function would only be possible for the interval (0,5].
>> implicitplot( fun(args(1)*exp(args(2))-args(2)*exp(-args(1)) - 1) , -4..4, -4..4, 7 ); show image
>> plot3d( Scaling=UnConstrained, [ Mode=Surface,[u, v, u^2*cos(v) + sin(u)], u=[-4.0,4.0],v=[-4.0,4.0], Color=[Height,[0.996109,0.164050,0.0], [0.992203,0.996109,0.0]], Style=[ColorPatches,AndMesh] ] ); show image
Unable to do in MuPAD 1.3
>> fieldplot( Axes = Origin, [ [-y^2, x^2], x = [-4, 4], y = [-4, 4], Grid = [30, 30] ] ); show image