I use Maxima for calculations. I solve a system of nonlinear equations using Newton's method (mnewton()). I get the solution in the form of a list:
[[φ2=5.921818183272879,s=5.155870949147037]]
How to get the numerical value of the first (φ2) and second (s) unknown. If I substitute:
x: roz1[1][2]$
I get that x is equal to: s=5.155870949147037
What to do to make x equal to a numerical value only: 5.155870949147037
(without s=).
My code:
Maxima code
I have two ideas. (1) You can call rhs to return the right-hand side of an equation (likewise lhs for the left-hand side). E.g. rhs(s = 123) returns 123.
(2) You can call assoc to find the value associated with s (or any variable) in the mnewton results. E.g. assoc('s, [a = 1, b = 2, s = 3, u = 5]) returns 3.
I like (2) better since it is not necessary to know where in the list is the one that you're interested in.
Related
I calculated a velocity vector and its module from the equations of motion of a point in wxMaxima:
x:3*sin(4*t);
y:2*cos(4*t);
r:[x,y];
v:diff(r,t,1);
v_mod:sqrt(v.v);
Now I would like to calculate the velocity for t=5. How can I do this? When I add (t) and := everywhere, like this:
x(t):=3*sin(4*t);
y(t):=2*cos(4*t);
r(t):=[x(t),y(t)];
v(t):=diff(r(t),t,1);
v_mod(t):=sqrt(v(t).v(t));
and then add this line at the end:
v_mod(5);
I get the following error:
diff: second argument must be a variable; found 5
What am I doing wrong here?
The problem is that when you say v(5), you're getting diff(<something>, 5) and Maxima is complaining about that.
Try v(t) := at(diff(r(u), u), u = t) -- i.e., differentiate wrt a dummy variable u, and then evaluate that derivative at u equal to the argument t.
There are other ways to go about it. If at doesn't work for you, we can try something else.
I'd like the example computation expression and values below to return 6. For some the numbers aren't yielding like I'd expect. What's the step I'm missing to get my result? Thanks!
type AddBuilder() =
let mutable x = 0
member _.Yield i = x <- x + i
member _.Zero() = 0
member _.Return() = x
let add = AddBuilder()
(* Compiler tells me that each of the numbers in add don't do anything
and suggests putting '|> ignore' in front of each *)
let result = add { 1; 2; 3 }
(* Currently the result is 0 *)
printfn "%i should be 6" result
Note: This is just for creating my own computation expression to expand my learning. Seq.sum would be a better approach. I'm open to the idea that this example completely misses the value of computation expressions and is no good for learning.
There is a lot wrong here.
First, let's start with mere mechanics.
In order for the Yield method to be called, the code inside the curly braces must use the yield keyword:
let result = add { yield 1; yield 2; yield 3 }
But now the compiler will complain that you also need a Combine method. See, the semantics of yield is that each of them produces a finished computation, a resulting value. And therefore, if you want to have more than one, you need some way to "glue" them together. This is what the Combine method does.
Since your computation builder doesn't actually produce any results, but instead mutates its internal variable, the ultimate result of the computation should be the value of that internal variable. So that's what Combine needs to return:
member _.Combine(a, b) = x
But now the compiler complains again: you need a Delay method. Delay is not strictly necessary, but it's required in order to mitigate performance pitfalls. When the computation consists of many "parts" (like in the case of multiple yields), it's often the case that some of them should be discarded. In these situation, it would be inefficient to evaluate all of them and then discard some. So the compiler inserts a call to Delay: it receives a function, which, when called, would evaluate a "part" of the computation, and Delay has the opportunity to put this function in some sort of deferred container, so that later Combine can decide which of those containers to discard and which to evaluate.
In your case, however, since the result of the computation doesn't matter (remember: you're not returning any results, you're just mutating the internal variable), Delay can just execute the function it receives to have it produce the side effects (which are - mutating the variable):
member _.Delay(f) = f ()
And now the computation finally compiles, and behold: its result is 6. This result comes from whatever Combine is returning. Try modifying it like this:
member _.Combine(a, b) = "foo"
Now suddenly the result of your computation becomes "foo".
And now, let's move on to semantics.
The above modifications will let your program compile and even produce expected result. However, I think you misunderstood the whole idea of the computation expressions in the first place.
The builder isn't supposed to have any internal state. Instead, its methods are supposed to manipulate complex values of some sort, some methods creating new values, some modifying existing ones. For example, the seq builder1 manipulates sequences. That's the type of values it handles. Different methods create new sequences (Yield) or transform them in some way (e.g. Combine), and the ultimate result is also a sequence.
In your case, it looks like the values that your builder needs to manipulate are numbers. And the ultimate result would also be a number.
So let's look at the methods' semantics.
The Yield method is supposed to create one of those values that you're manipulating. Since your values are numbers, that's what Yield should return:
member _.Yield x = x
The Combine method, as explained above, is supposed to combine two of such values that got created by different parts of the expression. In your case, since you want the ultimate result to be a sum, that's what Combine should do:
member _.Combine(a, b) = a + b
Finally, the Delay method should just execute the provided function. In your case, since your values are numbers, it doesn't make sense to discard any of them:
member _.Delay(f) = f()
And that's it! With these three methods, you can add numbers:
type AddBuilder() =
member _.Yield x = x
member _.Combine(a, b) = a + b
member _.Delay(f) = f ()
let add = AddBuilder()
let result = add { yield 1; yield 2; yield 3 }
I think numbers are not a very good example for learning about computation expressions, because numbers lack the inner structure that computation expressions are supposed to handle. Try instead creating a maybe builder to manipulate Option<'a> values.
Added bonus - there are already implementations you can find online and use for reference.
1 seq is not actually a computation expression. It predates computation expressions and is treated in a special way by the compiler. But good enough for examples and comparisons.
I am building a genetic algorithm that does a time series forecast in the symbolic regression analysis. I’m trying to get the algorithm to find an equation that will match the underlying trend of the data. (predict monthly beer sales)
The idea is to use lisp like expressions, which writes the equation in a tree. This allows for branch swapping in the crossover/mating stage.
5* (5 +5)
Written as:
X = '(mul 5 (add 5 5))'
Y = parser(X)
y = ['mul', 5, ['add', 5, 5]]
I want to know how to create an initial population set where the individuals represent different expressions automatically. Where there “fitness” is related to how well each equation matches the underlying trend.
For example, one individual could be: '(add 100 (mul x (sin (mul x 3))))'
Where x is time in months.
How do I automatically generate expressions for my population? I have no idea how to do this, any help would be very appreciated.
You can easily solve this problem with recursion and a random number generator random() which returns a (pseudo-)random float between 0 and 1. Here is some pseudocode:
randomExp() {
// Choose a function(like mul or add):
func = getRandomFunction() // Just choose one of your functions randomly.
arg1 = ""
rand1 = random()
// Choose the arguments. You may choose other percentages here depending how deep you want it to be and how many 'x' you want to have.
if(rand1 < 0.2)
arg1 = randomExp() // Here add a new expression
else if(rand1 < 0.5)
arg1 = "x"
else
arg1 = randomConstant() // Get a random constant in a predefined range.
// Do the same for the second argument:
arg2 = ""
…
…
// Put everything together and return it:
return "("+func+" "+arg1+" "+arg2+")"
}
You might want to also limit the recursion depth, as this might return you a theoretically infinitely long expression.
I want to make an arithmetic solver in Prolog that can have +,-,*,^ operations on numbers >= 2. It should also be possible to have a variable x in there. The input should be a prefix expression in a list.
I have made a program that parses an arithmetic expression in prefix format into a syntax tree. So that:
?- parse([+,+,2,9,*,3,x],Tree).
Tree = plus(plus(num(2), num(9)), mul(num(3), var(x))) .
(1) At this stage, I want to extend this program to be able to solve it for a given x value. This should be done by adding another predicate evaluate(Tree, Value, Solution) which given a value for the unknown x, calculates the solution.
Example:
?- parse([*, 2, ^, x, 3],Tree), evaluate(Ast, 2, Solution).
Tree = mul(num(2), pow(var(x), num(3))) ,
Solution = 16.
I'm not sure how to solve this problem due to my lack of Prolog skills, but I need a way of setting the var(x) to num(2) like in this example (because x = 2). Maybe member in Prolog can be used to do this. Then I have to solve it using perhaps is/2
Edit: My attempt to solving it. Getting error: 'Undefined procedure: evaluate/3 However, there are definitions for: evaluate/5'
evaluate(plus(A,B),Value,Sol) --> evaluate(A,AV,Sol), evaluate(B,BV,Sol), Value is AV+BV.
evaluate(mul(A,B),Value,Sol) --> evaluate(A,AV,Sol), evaluate(B,BV,Sol), Value is AV*BV.
evaluate(pow(A,B),Value,Sol) --> evaluate(A,AV,Sol), evaluate(B,BV,Sol), Value is AV^BV.
evaluate(num(Num),Value,Sol) --> number(Num).
evaluate(var(x),Value,Sol) --> number(Value).
(2) I'd also want to be able to express it in postfix form. Having a predicate postfixform(Tree, Postfixlist)
Example:
?- parse([+, *, 2, x, ^, x, 5 ],Tree), postfix(Tree,Postfix).
Tree = plus(mul(num(2), var(x)), pow(var(x), num(5))) ,
Postfix = [2, x, *, x, 5, ^, +].
Any help with (1) and (2) would be highly appreciated!
You don't need to use a grammar for this, as you are doing. You should use normal rules.
This is the pattern you need to follow.
evaluate(plus(A,B),Value,Sol) :-
evaluate(A, Value, A2),
evaluate(B, Value, B2),
Sol is A2+B2.
And
evaluate(num(X),_Value,Sol) :- Sol = X.
evaluate(var(x),Value,Sol) :- Sol = Value.
I am trying to solve a problem, for example I have a 4 point and each two point has a cost between them. Now I want to find a sequence of nodes which total cost would be less than a bound. I have written a code but it seems not working. The main problem is I have define a python function and trying to call it with in a constraint.
Here is my code: I have a function def getVal(n1,n2): where n1, n2 are Int Sort. The line Nodes = [ Int("n_%s" % (i)) for i in range(totalNodeNumber) ] defines 4 points as Int sort and when I am adding a constraint s.add(getVal(Nodes[0], Nodes[1]) + getVal(Nodes[1], Nodes[2]) < 100) then it calls getVal function immediately. But I want that, when Z3 will decide a value for Nodes[0], Nodes[1], Nodes[2], Nodes[3] then the function should be called for getting the cost between to points.
from z3 import *
import random
totalNodeNumber = 4
Nodes = [ Int("n_%s" % (i)) for i in range(totalNodeNumber) ]
def getVal(n1,n2):
# I need n1 and n2 values those assigned by Z3
cost = random.randint(1,20)
print cost
return IntVal(cost)
s = Solver()
#constraint: Each Nodes value should be distinct
nodes_index_distinct_constraint = Distinct(Nodes)
s.add(nodes_index_distinct_constraint)
#constraint: Each Nodes value should be between 0 and totalNodeNumber
def get_node_index_value_constraint(i):
return And(Nodes[i] >= 0, Nodes[i] < totalNodeNumber)
nodes_index_constraint = [ get_node_index_value_constraint(i) for i in range(totalNodeNumber)]
s.add(nodes_index_constraint)
#constraint: Problem with this constraint
# Here is the problem it's just called python getVal function twice without assiging Nodes[0],Nodes[1],Nodes[2] values
# But I want to implement that - Z3 will call python function during his decission making of variables
s.add(getVal(Nodes[0], Nodes[1]) + getVal(Nodes[1], Nodes[2]) + getVal(Nodes[2], Nodes[3]) < 100)
if s.check() == sat:
print "SAT"
print "Model: "
m = s.model()
nodeIndex = [ m.evaluate(Nodes[i]) for i in range(totalNodeNumber) ]
print nodeIndex
else:
print "UNSAT"
print "No solution found !!"
If this is not a right way to solve the problem then could you please tell me what would be other alternative way to solve it. Can I encode this kind of problem to find optimal sequence of way points using Z3 solver?
I don't understand what problem you need to solve. Definitely, the way getVal is formulated does not make sense. It does not use the arguments n1, n2. If you want to examine values produced by a model, then you do this after Z3 returns from a call to check().
I don't think you can use a python function in your SMT logic. What you could alternatively is define getVal as a Function like this
getVal = Function('getVal',IntSort(),IntSort(),IntSort())
And constraint the edge weights as
s.add(And(getVal(0,1)==1,getVal(1,2)==2,getVal(0,2)==3))
The first two input parameters of getVal represent the node ids and the last integer represents the weight.