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occurred Jun 15, 2012 at 22:36

clarification of variables.

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edited Jun 15, 2012 at 19:27

193
5

I am using a recursion algorithm developed by Migdal for Lattice Field Theory, and I have the following code:

(** The dimension of the representation **)
dim[r_] := 2 r + 1 
(**The trace of a product around the loop, also the sum of the eigenvalues **)
χ[r_] := Sin[dim[r] θ/2]/Sin[θ/2] 
(** The main equation in the paper I went out to 12 terms in the above sum **)
migdal[a_] := Sum[
   ((1/dim[r]) NIntegrate[(Sin[θ/2]^2)/(2 π) a χ[r], {θ, 0, 4 π}])^(λ^2) 
 * dim[r] χ[r], {r, 0, 3, 1/2}
]^(λ^(d - 2)) 

(** The λ used as expansion parameter **)
λ = 2^(1/10) 

d = 4 


l6 = Table[Table[Simplify[
 Re[NIntegrate[
  Log[Nest[
   migdal, 
   Exp[(9/1000) (χ[1/2] - dim[1/2]) - (3/100000) (χ[1] - dim[1]) 
       + 0 (χ[3/2] - dim[3/2])],
   i]] χ[j] Sin[θ/2]^2/(2 π), {θ, 0, 4 π}]]
], {i,0, 20}] , {j, 1/2, 1, 1/2}]

and it says that

NIntegrate::ncvb: "NIntegrate failed to converge to prescribed accuracy 
after 9 recursive bisections in θ near {θ} = {9.71911}. NIntegrate obtained
6.127043317150083`*^-15 and 1.1472584029987046`*^-16 for the integral and
error estimates."

I have tried to change the MinRecursion, on the first NIntegrate, but then the integral converges too slowly, and there are further errors. Mathematica makes a number of recommendations having to do with changing error estimates, however I don't understand what it is referring to.

In my code $a$ is where my Boltzmann factor goes, more or less, as an initial input, followed by subsequent recursion of that factor. The factor itself depends the trace, $\chi$, and so depends on $\theta$.

FYI the initial conditions that are variable in this are the 9/1000, 3/100 000, and 0. The first two numbers are initial points that are plotted against each other with the first number corresponding to the x-axis, and the second corresponding to the y-axis. I have been varying their range andwhere from the current numbers to (1.2, 0.1) respectively.

If someone would be so kind as to help me with these error messages I would be grateful.

Thanks,

I am using a recursion algorithm developed by Migdal for Lattice Field Theory, and I have the following code:

(** The dimension of the representation **)
dim[r_] := 2 r + 1 
(**The trace of a product around the loop, also the sum of the eigenvalues **)
χ[r_] := Sin[dim[r] θ/2]/Sin[θ/2] 
(** The main equation in the paper I went out to 12 terms in the above sum **)
migdal[a_] := Sum[
   ((1/dim[r]) NIntegrate[(Sin[θ/2]^2)/(2 π) a χ[r], {θ, 0, 4 π}])^(λ^2) 
 * dim[r] χ[r], {r, 0, 3, 1/2}
]^(λ^(d - 2)) 

(** The λ used as expansion parameter **)
λ = 2^(1/10) 

d = 4 


l6 = Table[Table[Simplify[
 Re[NIntegrate[
  Log[Nest[
   migdal, 
   Exp[(9/1000) (χ[1/2] - dim[1/2]) - (3/100000) (χ[1] - dim[1]) 
       + 0 (χ[3/2] - dim[3/2])],
   i]] χ[j] Sin[θ/2]^2/(2 π), {θ, 0, 4 π}]]
], {i,0, 20}] , {j, 1/2, 1, 1/2}]

and it says that

NIntegrate::ncvb: "NIntegrate failed to converge to prescribed accuracy 
after 9 recursive bisections in θ near {θ} = {9.71911}. NIntegrate obtained
6.127043317150083`*^-15 and 1.1472584029987046`*^-16 for the integral and
error estimates."

I have tried to change the MinRecursion, on the first NIntegrate, but then the integral converges too slowly, and there are further errors. Mathematica makes a number of recommendations having to do with changing error estimates, however I don't understand what it is referring to.

FYI the initial conditions that are variable in this are the 9/1000, 3/100 000, and 0. The first two numbers are initial points that are plotted against each other with the first number corresponding to the x-axis, and the second corresponding to the y-axis. I have been varying their range andwhere from the current numbers to (1.2, 0.1) respectively.

If someone would be so kind as to help me with these error messages I would be grateful.

Thanks,

I am using a recursion algorithm developed by Migdal for Lattice Field Theory, and I have the following code:

(** The dimension of the representation **)
dim[r_] := 2 r + 1 
(**The trace of a product around the loop, also the sum of the eigenvalues **)
χ[r_] := Sin[dim[r] θ/2]/Sin[θ/2] 
(** The main equation in the paper I went out to 12 terms in the above sum **)
migdal[a_] := Sum[
   ((1/dim[r]) NIntegrate[(Sin[θ/2]^2)/(2 π) a χ[r], {θ, 0, 4 π}])^(λ^2) 
 * dim[r] χ[r], {r, 0, 3, 1/2}
]^(λ^(d - 2)) 

(** The λ used as expansion parameter **)
λ = 2^(1/10) 

d = 4 


l6 = Table[Table[Simplify[
 Re[NIntegrate[
  Log[Nest[
   migdal, 
   Exp[(9/1000) (χ[1/2] - dim[1/2]) - (3/100000) (χ[1] - dim[1]) 
       + 0 (χ[3/2] - dim[3/2])],
   i]] χ[j] Sin[θ/2]^2/(2 π), {θ, 0, 4 π}]]
], {i,0, 20}] , {j, 1/2, 1, 1/2}]

and it says that

NIntegrate::ncvb: "NIntegrate failed to converge to prescribed accuracy 
after 9 recursive bisections in θ near {θ} = {9.71911}. NIntegrate obtained
6.127043317150083`*^-15 and 1.1472584029987046`*^-16 for the integral and
error estimates."

I have tried to change the MinRecursion, on the first NIntegrate, but then the integral converges too slowly, and there are further errors. Mathematica makes a number of recommendations having to do with changing error estimates, however I don't understand what it is referring to.

In my code $a$ is where my Boltzmann factor goes, more or less, as an initial input, followed by subsequent recursion of that factor. The factor itself depends the trace, $\chi$, and so depends on $\theta$.

FYI the initial conditions that are variable in this are the 9/1000, 3/100 000, and 0. The first two numbers are initial points that are plotted against each other with the first number corresponding to the x-axis, and the second corresponding to the y-axis. I have been varying their range andwhere from the current numbers to (1.2, 0.1) respectively.

If someone would be so kind as to help me with these error messages I would be grateful.

Thanks,

formatted code

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edited Jun 15, 2012 at 18:11

rcollyer

34.1k
7
94
194

I am using a recursion algorithm developed by Migdal for Lattice Field Theory, and I have the following code:

(** The dimension of the representation **)
dim[r_] := 2 r + 1  
(****The Thetrace dimensionof a product around the loop, also the sum of the representationeigenvalues **)
\[Chi][r_]χ[r_] := Sin[dim[r] \[Theta]θ/2]/Sin[\[Theta]Sin[θ/2]  
(**The trace** ofThe amain productequation aroundin the loop,paper alsoI thewent sumout ofto 12 terms in the eigenvaluesabove sum **)
migdal[a_] := Sum[
   ((1/dim[r]) NIntegrate[(Sin[\[Theta]Sin[θ/2]^2)/(2 \[Pi]π) a \[Chi][r]χ[r], {\[Theta]θ, 0, 4 \[Pi]π}])^(\[Lambda]^2 λ^2) 
 * dim[r] \[Chi][r]χ[r], {r, 0, 3, 1/2} 
]^(\[Lambda]^λ^(d - 2))  

(** The main equation in the paper I went out to 12 termsλ inused theas aboveexpansion sumparameter **)
 
\[Lambda]λ = 2^(1/10) (** The \[Lambda] used as expansion parameter **)

d = 4  


l6 = Table[Table[Simplify[Re[NIntegrate[Log[Nest[migdalTable[Table[Simplify[
 Re[NIntegrate[
  Log[Nest[
   migdal, 
   Exp[(9/1000) (\[Chi][1χ[1/2] - dim[1/2]) - (3/100000) (\[Chi][1]χ[1] - dim[1]) 
       + 0 (\[Chi][3χ[3/2] - dim[3/2])],
   i]] \[Chi][j]χ[j] Sin[\[Theta]Sin[θ/2]^2/(2 \[Pi]π), {\[Theta]θ, 0, 4 [Pi]π}]]]]]
], {i,0, 20}] , {j, 1/2, 1, 1/2}]

and it says that

NIntegrate::ncvb: "NIntegrate failed to converge to prescribed accuracy  
after 9 recursive bisections in \[Theta]θ near {\[Theta]θ} = {9.71911}. NIntegrate obtained  
6.127043317150083`*^-15 and 1.1472584029987046`*^-16 for the integral and  
error estimates."

I have tried to change the MinRecursion, on the first NIntegrate, but then the integral converges too slowly, and there are further errors. Mathematica makes a number of recommendations having to do with changing error estimates, however I don't understand what it is referring to.

FYI the initial conditions that are variable in this are the 9/1000, 3/100 000, and 0. The first two numbers are initial points that are plotted against each other with the first number corresponding to the x-axis, and the second corresponding to the y-axis. I have been varying their range andwhere from the current numbers to (1.2, 0.1) respectively.

If someone would be so kind as to help me with these error messages I would be grateful.

Thanks,

I am using a recursion algorithm developed by Migdal for Lattice Field Theory, and I have the following code:

dim[r_] := 2 r + 1 (** The dimension of the representation **)
\[Chi][r_] := Sin[dim[r] \[Theta]/2]/Sin[\[Theta]/2] (**The trace of a product around the loop, also the sum of the eigenvalues **)
migdal[a_] := Sum[((1/dim[r]) NIntegrate[(Sin[\[Theta]/2]^2)/(2 \[Pi]) a \[Chi][r], {\[Theta], 0, 4 \[Pi]}])^(\[Lambda]^2 ) dim[r] \[Chi][r], {r, 0, 3, 1/2}]^(\[Lambda]^(d - 2)) (** The main equation in the paper I went out to 12 terms in the above sum **)
 
\[Lambda] = 2^(1/10) (** The \[Lambda] used as expansion parameter **)

d = 4 

l6 = Table[Table[Simplify[Re[NIntegrate[Log[Nest[migdal, Exp[(9/1000) (\[Chi][1/2] - dim[1/2]) - (3/100000) (\[Chi][1] - dim[1]) + 0 (\[Chi][3/2] - dim[3/2])], i]] \[Chi][j] Sin[\[Theta]/2]^2/(2 \[Pi]), {\[Theta], 0, 4 [Pi]}]]], {i,0, 20}] , {j, 1/2, 1, 1/2}]

and it says that

NIntegrate::ncvb: "NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in \[Theta] near {\[Theta]} = {9.71911}. NIntegrate obtained 6.127043317150083`*^-15 and 1.1472584029987046`*^-16 for the integral and error estimates."

I have tried to change the MinRecursion, on the first NIntegrate, but then the integral converges too slowly, and there are further errors. Mathematica makes a number of recommendations having to do with changing error estimates, however I don't understand what it is referring to.

FYI the initial conditions that are variable in this are the 9/1000, 3/100 000, and 0. The first two numbers are initial points that are plotted against each other with the first number corresponding to the x-axis, and the second corresponding to the y-axis. I have been varying their range andwhere from the current numbers to (1.2, 0.1) respectively.

If someone would be so kind as to help me with these error messages I would be grateful.

Thanks,

I am using a recursion algorithm developed by Migdal for Lattice Field Theory, and I have the following code:

(** The dimension of the representation **)
dim[r_] := 2 r + 1  
(**The trace of a product around the loop, also the sum of the eigenvalues **)
χ[r_] := Sin[dim[r] θ/2]/Sin[θ/2]  
(** The main equation in the paper I went out to 12 terms in the above sum **)
migdal[a_] := Sum[
   ((1/dim[r]) NIntegrate[(Sin[θ/2]^2)/(2 π) a χ[r], {θ, 0, 4 π}])^(λ^2) 
 * dim[r] χ[r], {r, 0, 3, 1/2} 
]^(λ^(d - 2))  

(** The λ used as expansion parameter **)
λ = 2^(1/10) 

d = 4  


l6 = Table[Table[Simplify[
 Re[NIntegrate[
  Log[Nest[
   migdal, 
   Exp[(9/1000) (χ[1/2] - dim[1/2]) - (3/100000) (χ[1] - dim[1]) 
       + 0 (χ[3/2] - dim[3/2])],
   i]] χ[j] Sin[θ/2]^2/(2 π), {θ, 0, 4 π}]]
], {i,0, 20}] , {j, 1/2, 1, 1/2}]

and it says that

NIntegrate::ncvb: "NIntegrate failed to converge to prescribed accuracy  
after 9 recursive bisections in θ near {θ} = {9.71911}. NIntegrate obtained 
6.127043317150083`*^-15 and 1.1472584029987046`*^-16 for the integral and 
error estimates."

I have tried to change the MinRecursion, on the first NIntegrate, but then the integral converges too slowly, and there are further errors. Mathematica makes a number of recommendations having to do with changing error estimates, however I don't understand what it is referring to.

FYI the initial conditions that are variable in this are the 9/1000, 3/100 000, and 0. The first two numbers are initial points that are plotted against each other with the first number corresponding to the x-axis, and the second corresponding to the y-axis. I have been varying their range andwhere from the current numbers to (1.2, 0.1) respectively.

If someone would be so kind as to help me with these error messages I would be grateful.

Thanks,

Source Link

asked Jun 15, 2012 at 17:45

kηives

193
5

Error Interpretation in NIntegrate

I am using a recursion algorithm developed by Migdal for Lattice Field Theory, and I have the following code:

dim[r_] := 2 r + 1 (** The dimension of the representation **)
\[Chi][r_] := Sin[dim[r] \[Theta]/2]/Sin[\[Theta]/2] (**The trace of a product around the loop, also the sum of the eigenvalues **)
migdal[a_] := Sum[((1/dim[r]) NIntegrate[(Sin[\[Theta]/2]^2)/(2 \[Pi]) a \[Chi][r], {\[Theta], 0, 4 \[Pi]}])^(\[Lambda]^2 ) dim[r] \[Chi][r], {r, 0, 3, 1/2}]^(\[Lambda]^(d - 2)) (** The main equation in the paper I went out to 12 terms in the above sum **)

\[Lambda] = 2^(1/10) (** The \[Lambda] used as expansion parameter **)

d = 4 

l6 = Table[Table[Simplify[Re[NIntegrate[Log[Nest[migdal, Exp[(9/1000) (\[Chi][1/2] - dim[1/2]) - (3/100000) (\[Chi][1] - dim[1]) + 0 (\[Chi][3/2] - dim[3/2])], i]] \[Chi][j] Sin[\[Theta]/2]^2/(2 \[Pi]), {\[Theta], 0, 4 [Pi]}]]], {i,0, 20}] , {j, 1/2, 1, 1/2}]

and it says that

NIntegrate::ncvb: "NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in \[Theta] near {\[Theta]} = {9.71911}. NIntegrate obtained 6.127043317150083`*^-15 and 1.1472584029987046`*^-16 for the integral and error estimates."

I have tried to change the MinRecursion, on the first NIntegrate, but then the integral converges too slowly, and there are further errors. Mathematica makes a number of recommendations having to do with changing error estimates, however I don't understand what it is referring to.

FYI the initial conditions that are variable in this are the 9/1000, 3/100 000, and 0. The first two numbers are initial points that are plotted against each other with the first number corresponding to the x-axis, and the second corresponding to the y-axis. I have been varying their range andwhere from the current numbers to (1.2, 0.1) respectively.

If someone would be so kind as to help me with these error messages I would be grateful.

Thanks,

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Error Interpretation in NIntegrate