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I aim to calculate and preserve the results from the maximization of a function with two arguments and one exogenous parameter, when the maximum can not be derived (in closed form) by maximize. For instance, let
f[x_,y_,a_]=Max[0,Min[a-y,1-x-y]
be the objective function where a is positive. The maximization shall take place over [0,1]^2, therefore I set
m[a_]=Maximize[{f[x, y, a], 0 <= x <= 1 && 0 <= y <= 1 && 0 <= a}, {x,y}]
Obviously m can be evaluated at any point a and it is therefore possible to plot the maximizing x by employing
Plot[x /. m[a][[2]], {a, 0.01, 1}]
As I need to do several plots and further derivations containing the optimal solutions x and y (which of course are functions of a), i would like to preserve/save the results from the optimization for further use. Is there an elegant way to do this, or do I have to write some kind of loop to extract the values myself?
Now that I've seen the full text of your comment on my original comment, I suspect that you do understand the differences between Set and SetDelayed well enough. I think what you may be looking for is memoisation, sometimes implemented a bit like this;
f[x_,y_] := f[x,y] = Max[0,Min[a-y,1-x-y]]
When you evaluate, for example f[3,4] for the first time it will evaluate to the entire expression to the right of the :=. The rhs is the assignment f[3,4] = Max[0,Min[a-y,1-x-y]]. Next time you evaluate f[3,4] Mathematica already has a value for it so doesn't need to recompute it, it just recalls it. In this example the stored value would be Max[0,Min[a-4,-6]] of course.
I remain a little uncertain of what you are trying to do so this answer may not be any use to you at all.
Simple approach
results = Table[{x, y, a} /. m[a][[2]], {a, 0.01, 1, .01}]
ListPlot[{#[[3]], #[[1]]} & /# results, Joined -> True]
(The Set = is ok here so long as 'a' is not previosly defined )
If you want to utilise Plot[]s automatic evaluation take a look at Reap[]/Sow[]
{p, data} = Reap[Plot[x /. Sow[m[a]][[2]], {a, 0.01, 1}]];
Show[p]
(this takes a few minutes as the function output is a mess..).
hmm try this again: assuming you want x,y,a and the minimum value:
{p, data} = Reap[Plot[x /. Sow[{a, m[a]}][[2, 2]], {a, 0.01, .1}]];
Show[p]
results = {#[[1]], x /. #[[2, 2]], y /. #[[2, 2]], #[[2, 1]]} & /# data[[1]]
BTW Your function appears to be independent of x over some ranges which is why the plot is a mess..
I am trying to get Mathematica to approximate an integral that is a function of various parameters. I don't need it to be extremely precise -- the answer will be a fraction, and 5 digits would be nice, but I'd settle for as few as 2.
The problem is that there is a symbolic integral buried in the main integral, and I can't use NIntegrate on it since its symbolic.
F[x_, c_] := (1 - (1 - x)^c)^c;
a[n_, c_, x_] := F[a[n - 1, c, x], c];
a[0, c_, x_] = x;
MyIntegral[n_,c_] :=
NIntegrate[Integrate[(D[a[n,c,y],y]*y)/(1-a[n,c,x]),{y,x,1}],{x,0,1}]
Mathematica starts hanging when n is greater than 2 and c is greater than 3 or so (generally as both n and c get a little higher).
Are there any tricks for rewriting this expression so that it can be evaluated more easily? I've played with different WorkingPrecision and AccuracyGoal and PrecisionGoal options on the outer NIntegrate, but none of that helps the inner integral, which is where the problem is. In fact, for the higher values of n and c, I can't even get Mathematica to expand the inner derivative, i.e.
Expand[D[a[4,6,y],y]]
hangs.
I am using Mathematica 8 for Students.
If anyone has any tips for how I can get M. to approximate this, I would appreciate it.
Since you only want a numerical output (or that's what you'll get anyway), you can convert the symbolic integration into a numerical one using just NIntegrate as follows:
Clear[a,myIntegral]
a[n_Integer?Positive, c_Integer?Positive, x_] :=
a[n, c, x] = (1 - (1 - a[n - 1, c, x])^c)^c;
a[0, c_Integer, x_] = x;
myIntegral[n_, c_] :=
NIntegrate[D[a[n, c, y], y]*y/(1 - a[n, c, x]), {x, 0, 1}, {y, x, 1},
WorkingPrecision -> 200, PrecisionGoal -> 5]
This is much faster than performing the integration symbolically. Here's a comparison:
yoda:
myIntegral[2,2]//Timing
Out[1]= {0.088441, 0.647376595...}
myIntegral[5,2]//Timing
Out[2]= {1.10486, 0.587502888...}
rcollyer:
MyIntegral[2,2]//Timing
Out[3]= {1.0029, 0.647376}
MyIntegral[5,2]//Timing
Out[4]= {27.1697, 0.587503006...}
(* Obtained with WorkingPrecision->500, PrecisionGoal->5, MaxRecursion->20 *)
Jand's function has timings similar to rcollyer's. Of course, as you increase n, you will have to increase your WorkingPrecision way higher than this, as you've experienced in your previous question. Since you said you only need about 5 digits of precision, I've explicitly set PrecisionGoal to 5. You can change this as per your needs.
To codify the comments, I'd try the following. First, to eliminate infinite recursion with regards to the variable, n, I'd rewrite your functions as
F[x_, c_] := (1 - (1-x)^c)^c;
(* see note below *)
a[n_Integer?Positive, c_, x_] := F[a[n - 1, c, x], c];
a[0, c_, x_] = x;
that way n==0 will actually be a stopping point. The ?Positive form is a PatternTest, and useful for applying additional conditions to the parameters. I suspect the issue is that NIntegrate is re-evaluating the inner Integrate for every value of x, so I'd pull that evaluation out, like
MyIntegral[n_,c_] :=
With[{ int = Integrate[(D[a[n,c,y],y]*y)/(1-a[n,c,x]),{y,x,1}] },
NIntegrate[int,{x,0,1}]
]
where With is one of several scoping constructs specifically for creating local constants.
Your comments indicate that the inner integral takes a long time, have you tried simplifying the integrand as it is a derivative of a times a function of a? It seems like the result of a chain rule expansion to me.
Note: as per Yoda's suggestion in the comments, you can add a cacheing, or memoization, mechanism to a. Change its definition to
d:a[n_Integer?Positive, c_, x_] := d = F[a[n - 1, c, x], c];
The trick here is that in d:a[ ... ], d is a named pattern that is used again in d = F[...] cacheing the value of a for those particular parameter values.
Say I have a crazy function, f, defined like so:
util[x_, y_, c_] := 0.5*Log[c-x] + 0.5*Log[c-y]
cost[x_, y_, l_] := c /. First[NSolve[util[x, y, c+l] == Log[10+l], c]]
prof[x_, y_] := 0.01*Norm[{x,y}, 2]
liquid[x_, y_] := 0.01*Norm[{x,y}, 2]
f[x_, y_, a_, b_] := cost[a, b, liquid[x,y] + liquid[a-x, b-y]] - Max[a,b]
- cost[0,0,0] + prof[x,y] + liquid[x,y] + prof[a-x, b-y] + liquid[a-x, b-y]
Now I call NMinimize like this:
NMinimize[{f[50, 50, k, j], k >= 49, k <= 51, j >= 49, j <= 51}, {j, k}]
Which tells me this:
{-21.0465, {j -> 51., k -> 49.}}
But then if I actually check what f[50,50,49,51] is, it's this:
0.489033
Which is pretty different from the -21.0465 that NMinimize said.
Is this par for the course with NMinimize?
Floating point errors compounding or whatnot?
Any ideas for beating NMinimize (or some such function) into submission?
It certainly seems to be related to your function f not being restricted to numerical arguments, plus the symbolic preprocessing performed by NMinimize. Once you change the signature to
f[x_?NumericQ, y_?NumericQ, a_?NumericQ, b_?NumericQ]:=...
The result is as expected, although it takes considerably longer to get it.
EDIT
We can dig deeper to reveal the true reason. First, note that your f (the original one, args unrestricted) is quite a function:
In[1423]:= f[50,50,49.,51.]
Out[1423]= 0.489033
In[1392]:= f[50,50,k,j]/.{j->51.`,k->49.`}
Out[1392]= -21.0465
The real culprit is NSolve, which gives two ordered solutions:
In[1398]:= NSolve[util[x,y,c+l]==Log[10+l],c]
Out[1398]= {{c->0.5 (-2. l+1. x+1. y-2. Sqrt[100.+20. l+1. l^2+0.25 x^2-0.5 x y+0.25 y^2])},
{c->0.5 (-2. l+1. x+1. y+2. Sqrt[100.+20. l+1. l^2+0.25 x^2-0.5 x y+0.25 y^2])}}
The problem is, what is the ordering. It turns out to be different for symbolic and numeric arguments to NSolve, because in the latter case we don't have any symbols around. This can be seen as:
In[1399]:=
Block[{cost},
cost[x_,y_,l_]:=c/.Last[NSolve[util[x,y,c+l]==Log[10+l],c]];
f[50,50,k,j]/.{j->51.,k->49.}]
Out[1399]= 0.489033
So you really have to settle on what is the right ordering for you, and which solution you really want to pick.
I need to find the minimum of a function f(t) = int g(t,x) dx over [0,1]. What I did in mathematica is as follows:
f[t_] = NIntegrate[g[t,x],{x,-1,1}]
FindMinimum[f[t],{t,t0}]
However mathematica halts at the first try, because NIntegrate does not work with the symbolic t. It needs a specific value to evaluate. Although Plot[f[t],{t,0,1}] works perferctly, FindMinimum stops at the initial point.
I cannot replace NIntegrate by Integrate, because the function g is a bit complicated and if you type Integrate, mathematica just keep running...
Any way to get around it? Thanks!
Try this:
In[58]:= g[t_, x_] := t^3 - t + x^2
In[59]:= f[t_?NumericQ] := NIntegrate[g[t, x], {x, -1, 1}]
In[60]:= FindMinimum[f[t], {t, 1}]
Out[60]= {-0.103134, {t -> 0.57735}}
In[61]:= Plot[f[t], {t, 0, 1}]
Two relevant changes I made to your code:
Define f with := instead of with =. This effectively gives a definition for f "later", when the user of f has supplied the values of the arguments. See SetDelayed.
Define f with t_?NumericQ instead of t_. This says, t can be anything numeric (Pi, 7, 0, etc). But not anything non-numeric (t, x, "foo", etc).
An ounce of analysis...
You can get an exact answer and completely avoid the heavy lifting of the numerical integration, as long as Mathematica can do symbolic integration of g[t,x] w.r.t x and then symbolic differentiation w.r.t. t. A less trivial example with a more complicated g[t,x] including polynomial products in x and t:
g[t_, x_] := t^2 + (7*t*x - (x^3)/13)^2;
xMax = 1; xMin = -1; f[t_?NumericQ] := NIntegrate[g[t, x], {x, xMin, xMax}];
tMin = 0; tMax = 1;Plot[f[t], {t, tMin, tMax}];
tNumericAtMin = t /. FindMinimum[f[t], {t, tMax}][[2]];
dig[t_, x_] := D[Integrate[g[t, x], x], t];
Print["Differentiated integral is ", dig[t, x]];
digAtXMax = dig[t, x] /. x -> xMax; digAtXMin = dig[t, x] /. x -> xMin;
tSymbolicAtMin = Resolve[digAtXMax - digAtXMin == 0 && tMin ≤ t ≤ tMax, {t}];
Print["Exact: ", tSymbolicAtMin[[2]]];
Print["Numeric: ", tNumericAtMin];
Print["Difference: ", tSymbolicAtMin [[2]] - tNumericAtMin // N];
with the result:
⁃Graphics⁃
Differentiated integral is 2 t x + 98 t x^3 / 3 - 14 x^5 / 65
Exact: 21/3380
Numeric: 0.00621302
Difference: -3.01143 x 10^-9
Minimum of the function can be only at zero-points of it's derivate, so why to integrate in the first place?
You can use FindRoot or Solve to find roots of g
Then you can verify that points are really local minimums by checking derivates of g (it should be positive at that point).
Then you can NIntegrate to find minimum value of f - only one numerical integration!
I'm trying to figure out how to use Mathematica to solve systems of equations where some of the variables and coefficients are vectors. A simple example would be something like
where I know A, V, and the magnitude of P, and I have to solve for t and the direction of P. (Basically, given two rays A and B, where I know everything about A but only the origin and magnitude of B, figure out what the direction of B must be such that it intersects A.)
Now, I know how to solve this sort of thing by hand, but that's slow and error-prone, so I was hoping I could use Mathematica to speed things along and error-check me. However, I can't see how to get Mathematica to symbolically solve equations involving vectors like this.
I've looked in the VectorAnalysis package, without finding anything there that seems relevant; meanwhile the Linear Algebra package only seems to have a solver for linear systems (which this isn't, since I don't know t or P, just |P|).
I tried doing the simpleminded thing: expanding the vectors into their components (pretend they're 3D) and solving them as if I were trying to equate two parametric functions,
Solve[
{ Function[t, {Bx + Vx*t, By + Vy*t, Bz + Vz*t}][t] ==
Function[t, {Px*t, Py*t, Pz*t}][t],
Px^2 + Py^2 + Pz^2 == Q^2 } ,
{ t, Px, Py, Pz }
]
but the "solution" that spits out is a huge mess of coefficients and congestion. It also forces me to expand out each of the dimensions I feed it.
What I want is a nice symbolic solution in terms of dot products, cross products, and norms:
But I can't see how to tell Solve that some of the coefficients are vectors instead of scalars.
Is this possible? Can Mathematica give me symbolic solutions on vectors? Or should I just stick with No.2 Pencil technology?
(Just to be clear, I'm not interested in the solution to the particular equation at top -- I'm asking if I can use Mathematica to solve computational geometry problems like that generally without my having to express everything as an explicit matrix of {Ax, Ay, Az}, etc.)
With Mathematica 7.0.1.0
Clear[A, V, P];
A = {1, 2, 3};
V = {4, 5, 6};
P = {P1, P2, P3};
Solve[A + V t == P, P]
outputs:
{{P1 -> 1 + 4 t, P2 -> 2 + 5 t, P3 -> 3 (1 + 2 t)}}
Typing out P = {P1, P2, P3} can be annoying if the array or matrix is large.
Clear[A, V, PP, P];
A = {1, 2, 3};
V = {4, 5, 6};
PP = Array[P, 3];
Solve[A + V t == PP, PP]
outputs:
{{P[1] -> 1 + 4 t, P[2] -> 2 + 5 t, P[3] -> 3 (1 + 2 t)}}
Matrix vector inner product:
Clear[A, xx, bb];
A = {{1, 5}, {6, 7}};
xx = Array[x, 2];
bb = Array[b, 2];
Solve[A.xx == bb, xx]
outputs:
{{x[1] -> 1/23 (-7 b[1] + 5 b[2]), x[2] -> 1/23 (6 b[1] - b[2])}}
Matrix multiplication:
Clear[A, BB, d];
A = {{1, 5}, {6, 7}};
BB = Array[B, {2, 2}];
d = {{6, 7}, {8, 9}};
Solve[A.BB == d]
outputs:
{{B[1, 1] -> -(2/23), B[2, 1] -> 28/23, B[1, 2] -> -(4/23), B[2, 2] -> 33/23}}
The dot product has an infix notation built in just use a period for the dot.
I do not think the cross product does however. This is how you use the Notation package to make one. "X" will become our infix form of Cross. I suggest coping the example from the Notation, Symbolize and InfixNotation tutorial. Also use the Notation Palette which helps abstract away some of the Box syntax.
Clear[X]
Needs["Notation`"]
Notation[x_ X y_\[DoubleLongLeftRightArrow]Cross[x_, y_]]
Notation[NotationTemplateTag[
RowBox[{x_, , X, , y_, }]] \[DoubleLongLeftRightArrow]
NotationTemplateTag[RowBox[{ ,
RowBox[{Cross, [,
RowBox[{x_, ,, y_}], ]}]}]]]
{a, b, c} X {x, y, z}
outputs:
{-c y + b z, c x - a z, -b x + a y}
The above looks horrible but when using the Notation Palette it looks like:
Clear[X]
Needs["Notation`"]
Notation[x_ X y_\[DoubleLongLeftRightArrow]Cross[x_, y_]]
{a, b, c} X {x, y, z}
I have run into some quirks using the notation package in the past versions of mathematica so be careful.
I don't have a general solution for you by any means (MathForum may be the better way to go), but there are some tips that I can offer you. The first is to do the expansion of your vectors into components in a more systematic way. For instance, I would solve the equation you wrote as follows.
rawSol = With[{coords = {x, y, z}},
Solve[
Flatten[
{A[#] + V[#] t == P[#] t & /# coords,
Total[P[#]^2 & /# coords] == P^2}],
Flatten[{t, P /# coords}]]];
Then you can work with the rawSol variable more easily. Next, because you are referring the vector components in a uniform way (always matching the Mathematica pattern v_[x|y|z]), you can define rules that will aid in simplifying them. I played around a bit before coming up with the following rules:
vectorRules =
{forms___ + vec_[x]^2 + vec_[y]^2 + vec_[z]^2 :> forms + vec^2,
forms___ + c_. v1_[x]*v2_[x] + c_. v1_[y]*v2_[y] + c_. v1_[z]*v2_[z] :>
forms + c v1\[CenterDot]v2};
These rules will simplify the relationships for vector norms and dot products (cross-products are left as a likely painful exercise for the reader). EDIT: rcollyer pointed out that you can make c optional in the rule for dot products, so you only need two rules for norms and dot products.
With these rules, I was immediately able to simplify the solution for t into a form very close to yours:
In[3] := t /. rawSol //. vectorRules // Simplify // InputForm
Out[3] = {(A \[CenterDot] V - Sqrt[A^2*(P^2 - V^2) +
(A \[CenterDot] V)^2])/(P^2 - V^2),
(A \[CenterDot] V + Sqrt[A^2*(P^2 - V^2) +
(A \[CenterDot] V)^2])/(P^2 - V^2)}
Like I said, it's not a complete way of solving these kinds of problems by any means, but if you're careful about casting the problem into terms that are easy to work with from a pattern-matching and rule-replacement standpoint, you can go pretty far.
I've taken a somewhat different approach to this issue. I've made some definitions that return this output:
Patterns that are known to be vector quantities may be specified using vec[_], patterns that have an OverVector[] or OverHat[] wrapper (symbols with a vector or hat over them) are assumed to be vectors by default.
The definitions are experimental and should be treated as such, but they seem to work well. I expect to add to this over time.
Here are the definitions. The need to be pasted into a Mathematica Notebook cell and converted to StandardForm to see them properly.
Unprotect[vExpand,vExpand$,Cross,Plus,Times,CenterDot];
(* vec[pat] determines if pat is a vector quantity.
vec[pat] can be used to define patterns that should be treated as vectors.
Default: Patterns are assumed to be scalar unless otherwise defined *)
vec[_]:=False;
(* Symbols with a vector hat, or vector operations on vectors are assumed to be vectors *)
vec[OverVector[_]]:=True;
vec[OverHat[_]]:=True;
vec[u_?vec+v_?vec]:=True;
vec[u_?vec-v_?vec]:=True;
vec[u_?vec\[Cross]v_?vec]:=True;
vec[u_?VectorQ]:=True;
(* Placeholder for matrix types *)
mat[a_]:=False;
(* Anything not defined as a vector or matrix is a scalar *)
scal[x_]:=!(vec[x]\[Or]mat[x]);
scal[x_?scal+y_?scal]:=True;scal[x_?scal y_?scal]:=True;
(* Scalars times vectors are vectors *)
vec[a_?scal u_?vec]:=True;
mat[a_?scal m_?mat]:=True;
vExpand$[u_?vec\[Cross](v_?vec+w_?vec)]:=vExpand$[u\[Cross]v]+vExpand$[u\[Cross]w];
vExpand$[(u_?vec+v_?vec)\[Cross]w_?vec]:=vExpand$[u\[Cross]w]+vExpand$[v\[Cross]w];
vExpand$[u_?vec\[CenterDot](v_?vec+w_?vec)]:=vExpand$[u\[CenterDot]v]+vExpand$[u\[CenterDot]w];
vExpand$[(u_?vec+v_?vec)\[CenterDot]w_?vec]:=vExpand$[u\[CenterDot]w]+vExpand$[v\[CenterDot]w];
vExpand$[s_?scal (u_?vec\[Cross]v_?vec)]:=Expand[s] vExpand$[u\[Cross]v];
vExpand$[s_?scal (u_?vec\[CenterDot]v_?vec)]:=Expand[s] vExpand$[u\[CenterDot]v];
vExpand$[Plus[x__]]:=vExpand$/#Plus[x];
vExpand$[s_?scal,Plus[x__]]:=Expand[s](vExpand$/#Plus[x]);
vExpand$[Times[x__]]:=vExpand$/#Times[x];
vExpand[e_]:=e//.e:>Expand[vExpand$[e]]
(* Some simplification rules *)
(u_?vec\[Cross]u_?vec):=\!\(\*OverscriptBox["0", "\[RightVector]"]\);
(u_?vec+\!\(\*OverscriptBox["0", "\[RightVector]"]\)):=u;
0v_?vec:=\!\(\*OverscriptBox["0", "\[RightVector]"]\);
\!\(\*OverscriptBox["0", "\[RightVector]"]\)\[CenterDot]v_?vec:=0;
v_?vec\[CenterDot]\!\(\*OverscriptBox["0", "\[RightVector]"]\):=0;
(a_?scal u_?vec)\[Cross]v_?vec :=a u\[Cross]v;u_?vec\[Cross](a_?scal v_?vec ):=a u\[Cross]v;
(a_?scal u_?vec)\[CenterDot]v_?vec :=a u\[CenterDot]v;
u_?vec\[CenterDot](a_?scal v_?vec) :=a u\[CenterDot]v;
(* Stealing behavior from Dot *)
Attributes[CenterDot]=Attributes[Dot];
Protect[vExpand,vExpand$,Cross,Plus,Times,CenterDot];