Consider the following function, given in recursive manner:
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N0(t(t<0) or (t>1))==0
Type: Void
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N0(t(t>=0) and (t<=1))==1
Type: Void
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N(t,i,0)==N0(ti)
Type: Void
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N(t,i,pp>0)==(ti)/p*N(t,i,p1)+(i+1t)/p*N(t,i+1,p1)
Type: Void
This is a way to create (uniform) bsplines. Now try to differentiate
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D(N(t,0,3),t)
There are 3 exposed and 1 unexposed library operations named <
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op <
to learn more about the available operations. Perhaps
packagecalling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named <
with argument type(s)
Polynomial(Integer)
NonNegativeInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
FriCAS will attempt to step through and interpret the code.
There are 3 exposed and 1 unexposed library operations named <
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op <
to learn more about the available operations. Perhaps
packagecalling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named <
with argument type(s)
Polynomial(Integer)
NonNegativeInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
FriCAS will attempt to step through and interpret the code.
There are 3 exposed and 1 unexposed library operations named <
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op <
to learn more about the available operations. Perhaps
packagecalling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named <
with argument type(s)
Polynomial(Integer)
NonNegativeInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Yack!!! This is obviously wrong! The map
is
continuous
and
is not constant, despite what Axiom seems to claim here.
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N(t,0,3)
There are 3 exposed and 1 unexposed library operations named <
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op <
to learn more about the available operations. Perhaps
packagecalling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named <
with argument type(s)
Polynomial(Integer)
NonNegativeInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
FriCAS will attempt to step through and interpret the code.
There are 3 exposed and 1 unexposed library operations named <
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op <
to learn more about the available operations. Perhaps
packagecalling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named <
with argument type(s)
Polynomial(Integer)
NonNegativeInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
On the other hand the function is not constant.
See this:
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N0(t(t<0) or (t>1))==0;
Compiled code for N0 has been cleared.
Compiled code for N has been cleared.
1 old definition(s) deleted for function or rule N0
Type: Void
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N0(t(t>=0) and (t<=1))==1;
1 old definition(s) deleted for function or rule N0
Type: Void
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N(t,i,0)==N0(ti);
1 old definition(s) deleted for function or rule N
Type: Void
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N(t,i,pp>0)==(ti)/p*N(t,i,p1)+(i+1t)/p*N(t,i+1,p1);
1 old definition(s) deleted for function or rule N
Type: Void
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for x in 5..15 repeat output N(x/10,0,3)
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Compiling function N0 with type Fraction(Integer) >
NonNegativeInteger
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Compiling function N with type (Fraction(Integer), Integer, Integer)
> Fraction(Integer)
0
0
0
0
0
0
1

6000
1

750
9

2000
4

375
1

48
9

250
343

6000
32

375
243

2000
1

6
133
 
6000
43
 
750
217
 
2000
67
 
375
13
 
48
Type: Void
Drawing the plot (unfortunately not available here) would
show it even more clearly.
But in D(N(t,0,3),t)
you are not calling the function N
with numeric parameters. In N(t,0,3)
the type of t is
Variable t
. Ultimately N(t,0,3)=0
because of your function
definition N0(t(t<0) or (t>1))==0
. This is 0
because
t>1
is true
when t
is of type Variable t
. You can
see why if you use the option )set message bottomup on
to
see the mode map selection
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)set message bottomup on
t>1
Function Selection for >
Arguments: (VARIABLE(t), PI)
> no appropriate > found in Variable(t)
> no appropriate > found in PositiveInteger
> no appropriate > found in Symbol
> no appropriate > found in Integer
> no appropriate > found in Variable(t)
> no appropriate > found in PositiveInteger
> no appropriate > found in Symbol
> no appropriate > found in Integer
Modemaps from Associated Packages
no modemaps
Remaining General Modemaps
[1] (D,D) > Boolean from D if D has PORDER
> no function > found for arguments (VARIABLE(t), PI)
Function Selection for >
Arguments: (VARIABLE(t), NNI)
> no appropriate > found in Variable(t)
> no appropriate > found in NonNegativeInteger
> no appropriate > found in Symbol
> no appropriate > found in Integer
> no appropriate > found in Variable(t)
> no appropriate > found in NonNegativeInteger
> no appropriate > found in Symbol
> no appropriate > found in Integer
Modemaps from Associated Packages
no modemaps
Remaining General Modemaps
[1] (D,D) > Boolean from D if D has PORDER
> no function > found for arguments (VARIABLE(t), NNI)
Function Selection for >
Arguments: (VARIABLE(t), INT)
> no appropriate > found in Variable(t)
> no appropriate > found in Integer
> no appropriate > found in Symbol
> no appropriate > found in Variable(t)
> no appropriate > found in Integer
> no appropriate > found in Symbol
Modemaps from Associated Packages
no modemaps
Remaining General Modemaps
[1] (D,D) > Boolean from D if D has PORDER
> no function > found for arguments (VARIABLE(t), INT)
Function Selection for >
Arguments: (VARIABLE(t), SINT)
> no appropriate > found in Variable(t)
> no appropriate > found in SingleInteger
> no appropriate > found in Symbol
> no appropriate > found in Variable(t)
> no appropriate > found in SingleInteger
> no appropriate > found in Symbol
Modemaps from Associated Packages
no modemaps
Remaining General Modemaps
[1] (D,D) > Boolean from D if D has PORDER
> no function > found for arguments (VARIABLE(t), SINT)
There are 1 exposed and 2 unexposed library operations named >
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op >
to learn more about the available operations. Perhaps
packagecalling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named >
with argument type(s)
Variable(t)
PositiveInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Axiom interprets both t
and 1
as being of type POLY INT
and the function >
is defined by the lexical ordering of the
polynomials.
This result is counterintuitive, but once you understand why
Axiom gives this result then you will be in a good position to
understand the rest of Axiom's type system!
It is possible to write the function N0 so that it returns the
desired result. See ExampleSolution1?.