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How to count the number of function evaluations in NIntegrate

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2

$begingroup$

I expect the following code to count the number of function calls in NIntegrate.

i = 0;
f[x_] := (i += 1; Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]])
Print[NIntegrate[f[x], x, -2, 1]] 
Print[i]

However, the output is

0.0901049
2

which means only 2 function calls in NIntegrate. It's hard to believe converged result can be obtained by merely two function calls. What's happening here?

numerical-integration

share|improve this question

asked 8 hours ago

Chong WangChong Wang

25422 silver badges88 bronze badges

$endgroup$

add a comment
 | 

2

$begingroup$

I expect the following code to count the number of function calls in NIntegrate.

i = 0;
f[x_] := (i += 1; Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]])
Print[NIntegrate[f[x], x, -2, 1]] 
Print[i]

However, the output is

0.0901049
2

which means only 2 function calls in NIntegrate. It's hard to believe converged result can be obtained by merely two function calls. What's happening here?

numerical-integration

share|improve this question

asked 8 hours ago

Chong WangChong Wang

25422 silver badges88 bronze badges

$endgroup$

add a comment
 | 

$begingroup$

I expect the following code to count the number of function calls in NIntegrate.

i = 0;
f[x_] := (i += 1; Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]])
Print[NIntegrate[f[x], x, -2, 1]] 
Print[i]

However, the output is

0.0901049
2

which means only 2 function calls in NIntegrate. It's hard to believe converged result can be obtained by merely two function calls. What's happening here?

numerical-integration

share|improve this question

asked 8 hours ago

Chong WangChong Wang

25422 silver badges88 bronze badges

$endgroup$

i = 0;
f[x_] := (i += 1; Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]])
Print[NIntegrate[f[x], x, -2, 1]] 
Print[i]

0.0901049
2

share|improve this question

asked 8 hours ago

Chong WangChong Wang

25422 silver badges88 bronze badges

share|improve this question

asked 8 hours ago

Chong WangChong Wang

25422 silver badges88 bronze badges

asked 8 hours ago

Chong WangChong Wang

25422 silver badges88 bronze badges

asked 8 hours ago

Chong WangChong Wang

25422 silver badges88 bronze badges

2 Answers
2

active

oldest

votes

2

$begingroup$

Try the option EvaluationMonitor

Block[k = 0, NIntegrate[f[x], x, -2, 1, EvaluationMonitor :> k++], k]

0.0901049, 121

Without using EvaluationMonitor you can do

ClearAll[f, ff]
f[x_] := Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]]

i = 0;
ff[y_?NumberQ] := Block[x = y, i++; f[x]]

NIntegrate[ff[x], x, -2, 1], i

0.0901049, 121

share|improve this answer

edited 7 hours ago

answered 8 hours ago

kglrkglr

217k1010 gold badges247247 silver badges497497 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yeah, that works. But what's wrong with my approach?
  $endgroup$
  – Chong Wang
  8 hours ago
  

  
  
  
  

add a comment
 | 

2

$begingroup$

A very simple modification (adding ?NumericQ) achieves the result you expected.

i = 0;
f[x_?NumericQ] := (i += 1; Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]])
Print[NIntegrate[f[x], x, -2, 1]]
Print[i]

(* 0.0901049*)

(*122*)

The issue is that NIntegrate tries to evaluate f[x] symbolically. In your version, it is called only once and is replaced by Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]. In my version, the f[x] is returned unchanged until specific numerical values are given to x.

share|improve this answer

answered 7 hours ago

mikadomikado

7,63711 gold badge99 silver badges2929 bronze badges

$endgroup$

add a comment
 | 

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2

$begingroup$

Try the option EvaluationMonitor

Block[k = 0, NIntegrate[f[x], x, -2, 1, EvaluationMonitor :> k++], k]

0.0901049, 121

Without using EvaluationMonitor you can do

ClearAll[f, ff]
f[x_] := Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]]

i = 0;
ff[y_?NumberQ] := Block[x = y, i++; f[x]]

NIntegrate[ff[x], x, -2, 1], i

0.0901049, 121

share|improve this answer

edited 7 hours ago

answered 8 hours ago

kglrkglr

217k1010 gold badges247247 silver badges497497 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yeah, that works. But what's wrong with my approach?
  $endgroup$
  – Chong Wang
  8 hours ago
  

  
  
  
  

add a comment
 | 

2

$begingroup$

Try the option EvaluationMonitor

Block[k = 0, NIntegrate[f[x], x, -2, 1, EvaluationMonitor :> k++], k]

0.0901049, 121

Without using EvaluationMonitor you can do

ClearAll[f, ff]
f[x_] := Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]]

i = 0;
ff[y_?NumberQ] := Block[x = y, i++; f[x]]

NIntegrate[ff[x], x, -2, 1], i

0.0901049, 121

share|improve this answer

edited 7 hours ago

answered 8 hours ago

kglrkglr

217k1010 gold badges247247 silver badges497497 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yeah, that works. But what's wrong with my approach?
  $endgroup$
  – Chong Wang
  8 hours ago
  

  
  
  
  

add a comment
 | 

$begingroup$

Try the option EvaluationMonitor

Block[k = 0, NIntegrate[f[x], x, -2, 1, EvaluationMonitor :> k++], k]

0.0901049, 121

Without using EvaluationMonitor you can do

ClearAll[f, ff]
f[x_] := Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]]

i = 0;
ff[y_?NumberQ] := Block[x = y, i++; f[x]]

NIntegrate[ff[x], x, -2, 1], i

0.0901049, 121

share|improve this answer

edited 7 hours ago

answered 8 hours ago

kglrkglr

217k1010 gold badges247247 silver badges497497 bronze badges

$endgroup$

Block[k = 0, NIntegrate[f[x], x, -2, 1, EvaluationMonitor :> k++], k]

ClearAll[f, ff]
f[x_] := Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]]

i = 0;
ff[y_?NumberQ] := Block[x = y, i++; f[x]]

NIntegrate[ff[x], x, -2, 1], i

share|improve this answer

edited 7 hours ago

answered 8 hours ago

kglrkglr

217k1010 gold badges247247 silver badges497497 bronze badges

answered 8 hours ago

kglrkglr

217k1010 gold badges247247 silver badges497497 bronze badges

answered 8 hours ago

kglrkglr

217k1010 gold badges247247 silver badges497497 bronze badges

2

$begingroup$

A very simple modification (adding ?NumericQ) achieves the result you expected.

i = 0;
f[x_?NumericQ] := (i += 1; Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]])
Print[NIntegrate[f[x], x, -2, 1]]
Print[i]

(* 0.0901049*)

(*122*)

The issue is that NIntegrate tries to evaluate f[x] symbolically. In your version, it is called only once and is replaced by Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]. In my version, the f[x] is returned unchanged until specific numerical values are given to x.

share|improve this answer

answered 7 hours ago

mikadomikado

7,63711 gold badge99 silver badges2929 bronze badges

$endgroup$

add a comment
 | 

2

$begingroup$

A very simple modification (adding ?NumericQ) achieves the result you expected.

i = 0;
f[x_?NumericQ] := (i += 1; Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]])
Print[NIntegrate[f[x], x, -2, 1]]
Print[i]

(* 0.0901049*)

(*122*)

The issue is that NIntegrate tries to evaluate f[x] symbolically. In your version, it is called only once and is replaced by Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]. In my version, the f[x] is returned unchanged until specific numerical values are given to x.

share|improve this answer

answered 7 hours ago

mikadomikado

7,63711 gold badge99 silver badges2929 bronze badges

$endgroup$

add a comment
 | 

$begingroup$

A very simple modification (adding ?NumericQ) achieves the result you expected.

i = 0;
f[x_?NumericQ] := (i += 1; Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]])
Print[NIntegrate[f[x], x, -2, 1]]
Print[i]

(* 0.0901049*)

(*122*)

The issue is that NIntegrate tries to evaluate f[x] symbolically. In your version, it is called only once and is replaced by Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]. In my version, the f[x] is returned unchanged until specific numerical values are given to x.

share|improve this answer

answered 7 hours ago

mikadomikado

7,63711 gold badge99 silver badges2929 bronze badges

$endgroup$

i = 0;
f[x_?NumericQ] := (i += 1; Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]])
Print[NIntegrate[f[x], x, -2, 1]]
Print[i]

(* 0.0901049*)

(*122*)

share|improve this answer

answered 7 hours ago

mikadomikado

7,63711 gold badge99 silver badges2929 bronze badges

answered 7 hours ago

mikadomikado

7,63711 gold badge99 silver badges2929 bronze badges

answered 7 hours ago

mikadomikado

7,63711 gold badge99 silver badges2929 bronze badges