Why is this code so slow?












2












$begingroup$


This code for the first five iterations the speed is okay , but after that the speed is very slow, I cannot understand what is wrong with this? Would you please help me fix it?



  Clear[A, r, x, s, e]

s := 0.3405

e := 1.6539*10^-21

u[0] := 0.

u[1] := 0.1



A[r_] := A[r] = 

  Piecewise[{{r - 2.5 s - 48*e *s^12*r^-13 + 24*e*s^6*r^-7, 

     r > 2.5 s}, {-48*e*s^12*r^-13 + 24*e*s^6*r^-7, 

     s [LessSlantEqual] r [LessSlantEqual] 2.5 s}, {r - s - 

      24*e*s^-1, r < s}}]

For[i = 2, i < 101, 

 i++, { u[i_] := 

   x /. FindRoot[

     u[i - 1] + 

       1/(i^2 (u[i - 1] - u[i - 2])^2) (u[i - 1] - u[i - 2]) - 

       0.9 A[x] == x , {x, 1.}]; Print[u[i]]}]





equation-solving iteration







share|improve this question









$endgroup$

















    2












    $begingroup$


    This code for the first five iterations the speed is okay , but after that the speed is very slow, I cannot understand what is wrong with this? Would you please help me fix it?



      Clear[A, r, x, s, e]
    
s := 0.3405
    
e := 1.6539*10^-21
    
u[0] := 0.
    
u[1] := 0.1
    

    
A[r_] := A[r] = 
    
  Piecewise[{{r - 2.5 s - 48*e *s^12*r^-13 + 24*e*s^6*r^-7, 
    
     r > 2.5 s}, {-48*e*s^12*r^-13 + 24*e*s^6*r^-7, 
    
     s [LessSlantEqual] r [LessSlantEqual] 2.5 s}, {r - s - 
    
      24*e*s^-1, r < s}}]
    
For[i = 2, i < 101, 
    
 i++, { u[i_] := 
    
   x /. FindRoot[
    
     u[i - 1] + 
    
       1/(i^2 (u[i - 1] - u[i - 2])^2) (u[i - 1] - u[i - 2]) - 
    
       0.9 A[x] == x , {x, 1.}]; Print[u[i]]}]





    equation-solving iteration







    share|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      This code for the first five iterations the speed is okay , but after that the speed is very slow, I cannot understand what is wrong with this? Would you please help me fix it?



        Clear[A, r, x, s, e]
      
s := 0.3405
      
e := 1.6539*10^-21
      
u[0] := 0.
      
u[1] := 0.1
      

      
A[r_] := A[r] = 
      
  Piecewise[{{r - 2.5 s - 48*e *s^12*r^-13 + 24*e*s^6*r^-7, 
      
     r > 2.5 s}, {-48*e*s^12*r^-13 + 24*e*s^6*r^-7, 
      
     s [LessSlantEqual] r [LessSlantEqual] 2.5 s}, {r - s - 
      
      24*e*s^-1, r < s}}]
      
For[i = 2, i < 101, 
      
 i++, { u[i_] := 
      
   x /. FindRoot[
      
     u[i - 1] + 
      
       1/(i^2 (u[i - 1] - u[i - 2])^2) (u[i - 1] - u[i - 2]) - 
      
       0.9 A[x] == x , {x, 1.}]; Print[u[i]]}]





      equation-solving iteration







      share|improve this question









      $endgroup$




      This code for the first five iterations the speed is okay , but after that the speed is very slow, I cannot understand what is wrong with this? Would you please help me fix it?



        Clear[A, r, x, s, e]
      
s := 0.3405
      
e := 1.6539*10^-21
      
u[0] := 0.
      
u[1] := 0.1
      

      
A[r_] := A[r] = 
      
  Piecewise[{{r - 2.5 s - 48*e *s^12*r^-13 + 24*e*s^6*r^-7, 
      
     r > 2.5 s}, {-48*e*s^12*r^-13 + 24*e*s^6*r^-7, 
      
     s [LessSlantEqual] r [LessSlantEqual] 2.5 s}, {r - s - 
      
      24*e*s^-1, r < s}}]
      
For[i = 2, i < 101, 
      
 i++, { u[i_] := 
      
   x /. FindRoot[
      
     u[i - 1] + 
      
       1/(i^2 (u[i - 1] - u[i - 2])^2) (u[i - 1] - u[i - 2]) - 
      
       0.9 A[x] == x , {x, 1.}]; Print[u[i]]}]





      equation-solving iteration




      equation-solving iteration






      share|improve this question













      share|improve this question











      share|improve this question




      share|improve this question










      asked 2 hours ago









      morapimorapi

      204




      204






















          1 Answer
          1






          active

          oldest

          votes


















          5












          $begingroup$

          I recommend you learn the distinction between immediate (=) and delayed (:=) assignments. They make the difference between slow and fast code here. Start with this tutorial or this book chapter, then look at memoization.



          s = 0.3405;
          
e = 1.6539*10^-21;
          
u[0] = 0.;
          
u[1] = 0.1;
          

          
A[r_] = Piecewise[{{r - 2.5 s - 48*e*s^12*r^-13 + 24*e*s^6*r^-7, r > 2.5 s},
          
                   {-48*e*s^12*r^-13 + 24*e*s^6*r^-7, s <= r <= 2.5 s},
          
                   {r - s - 24*e*s^-1, r < s}}];
          

          
u[i_] := u[i] = x /. FindRoot[
          
  u[i - 1] + 1/(i^2 (u[i - 1] - u[i - 2])^2) (u[i - 1] - u[i - 2]) - 0.9 A[x] == x, {x, 1.}]
          

          
Array[u, 100]



          {0.1, 1.77164, 1.37065, 1.04259, 0.887781, 0.708344, 0.59461,
          0.457228, 0.367364, 0.296071, 0.256104, 0.20463, 0.208487, 1.20917,
          1.04197, 0.939331, 0.879865, 0.827963, 0.774591, 0.72775, 0.67934,
          0.63666, 0.592369, 0.553172, 0.512352, 0.476112, 0.438261, 0.404563,
          0.369277, 0.339073, 0.321616, 0.301118, 0.296195, 0.224688, 0.273538,
          0.31357, 0.33593, 0.366902, 0.38813, 0.417572, 0.437777, 0.465834,
          0.48511, 0.511907, 0.530336, 0.55598, 0.573633, 0.598219, 0.615159,
          0.638772, 0.655054, 0.677768, 0.693441, 0.715321, 0.73043, 0.751535,
          0.766118, 0.786503, 0.800596, 0.820306, 0.833941, 0.852182, 0.85901,
          0.874152, 0.871531, 0.78396, 0.781416, 0.696402, 0.693931, 0.611329,
          0.608927, 0.528603, 0.526267, 0.448099, 0.445825, 0.369701, 0.367485,
          0.315658, 0.325798, 0.341207, 0.351098, 0.366134, 0.375788, 0.390468,
          0.399897, 0.414237, 0.42345, 0.437466, 0.446473, 0.46018, 0.46899,
          0.4824, 0.491022, 0.504149, 0.51259, 0.525444, 0.533712, 0.546306,
          0.554408, 0.56675}




          (takes about 5 seconds)



          Alternatively, use



          Table[u[i], {i, 1, 100}]


          (same result). Your combination of For and Print shows the results but doesn't let you keep using them for more calculations.






          share|improve this answer











          $endgroup$













          • $begingroup$
            thank you very much. I really appreciate it.
            $endgroup$
            – morapi
            5 mins ago












          Your Answer





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          active

          oldest

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          5












          $begingroup$

          I recommend you learn the distinction between immediate (=) and delayed (:=) assignments. They make the difference between slow and fast code here. Start with this tutorial or this book chapter, then look at memoization.



          s = 0.3405;
          
e = 1.6539*10^-21;
          
u[0] = 0.;
          
u[1] = 0.1;
          

          
A[r_] = Piecewise[{{r - 2.5 s - 48*e*s^12*r^-13 + 24*e*s^6*r^-7, r > 2.5 s},
          
                   {-48*e*s^12*r^-13 + 24*e*s^6*r^-7, s <= r <= 2.5 s},
          
                   {r - s - 24*e*s^-1, r < s}}];
          

          
u[i_] := u[i] = x /. FindRoot[
          
  u[i - 1] + 1/(i^2 (u[i - 1] - u[i - 2])^2) (u[i - 1] - u[i - 2]) - 0.9 A[x] == x, {x, 1.}]
          

          
Array[u, 100]



          {0.1, 1.77164, 1.37065, 1.04259, 0.887781, 0.708344, 0.59461,
          0.457228, 0.367364, 0.296071, 0.256104, 0.20463, 0.208487, 1.20917,
          1.04197, 0.939331, 0.879865, 0.827963, 0.774591, 0.72775, 0.67934,
          0.63666, 0.592369, 0.553172, 0.512352, 0.476112, 0.438261, 0.404563,
          0.369277, 0.339073, 0.321616, 0.301118, 0.296195, 0.224688, 0.273538,
          0.31357, 0.33593, 0.366902, 0.38813, 0.417572, 0.437777, 0.465834,
          0.48511, 0.511907, 0.530336, 0.55598, 0.573633, 0.598219, 0.615159,
          0.638772, 0.655054, 0.677768, 0.693441, 0.715321, 0.73043, 0.751535,
          0.766118, 0.786503, 0.800596, 0.820306, 0.833941, 0.852182, 0.85901,
          0.874152, 0.871531, 0.78396, 0.781416, 0.696402, 0.693931, 0.611329,
          0.608927, 0.528603, 0.526267, 0.448099, 0.445825, 0.369701, 0.367485,
          0.315658, 0.325798, 0.341207, 0.351098, 0.366134, 0.375788, 0.390468,
          0.399897, 0.414237, 0.42345, 0.437466, 0.446473, 0.46018, 0.46899,
          0.4824, 0.491022, 0.504149, 0.51259, 0.525444, 0.533712, 0.546306,
          0.554408, 0.56675}




          (takes about 5 seconds)



          Alternatively, use



          Table[u[i], {i, 1, 100}]


          (same result). Your combination of For and Print shows the results but doesn't let you keep using them for more calculations.






          share|improve this answer











          $endgroup$













          • $begingroup$
            thank you very much. I really appreciate it.
            $endgroup$
            – morapi
            5 mins ago
















          5












          $begingroup$

          I recommend you learn the distinction between immediate (=) and delayed (:=) assignments. They make the difference between slow and fast code here. Start with this tutorial or this book chapter, then look at memoization.



          s = 0.3405;
          
e = 1.6539*10^-21;
          
u[0] = 0.;
          
u[1] = 0.1;
          

          
A[r_] = Piecewise[{{r - 2.5 s - 48*e*s^12*r^-13 + 24*e*s^6*r^-7, r > 2.5 s},
          
                   {-48*e*s^12*r^-13 + 24*e*s^6*r^-7, s <= r <= 2.5 s},
          
                   {r - s - 24*e*s^-1, r < s}}];
          

          
u[i_] := u[i] = x /. FindRoot[
          
  u[i - 1] + 1/(i^2 (u[i - 1] - u[i - 2])^2) (u[i - 1] - u[i - 2]) - 0.9 A[x] == x, {x, 1.}]
          

          
Array[u, 100]



          {0.1, 1.77164, 1.37065, 1.04259, 0.887781, 0.708344, 0.59461,
          0.457228, 0.367364, 0.296071, 0.256104, 0.20463, 0.208487, 1.20917,
          1.04197, 0.939331, 0.879865, 0.827963, 0.774591, 0.72775, 0.67934,
          0.63666, 0.592369, 0.553172, 0.512352, 0.476112, 0.438261, 0.404563,
          0.369277, 0.339073, 0.321616, 0.301118, 0.296195, 0.224688, 0.273538,
          0.31357, 0.33593, 0.366902, 0.38813, 0.417572, 0.437777, 0.465834,
          0.48511, 0.511907, 0.530336, 0.55598, 0.573633, 0.598219, 0.615159,
          0.638772, 0.655054, 0.677768, 0.693441, 0.715321, 0.73043, 0.751535,
          0.766118, 0.786503, 0.800596, 0.820306, 0.833941, 0.852182, 0.85901,
          0.874152, 0.871531, 0.78396, 0.781416, 0.696402, 0.693931, 0.611329,
          0.608927, 0.528603, 0.526267, 0.448099, 0.445825, 0.369701, 0.367485,
          0.315658, 0.325798, 0.341207, 0.351098, 0.366134, 0.375788, 0.390468,
          0.399897, 0.414237, 0.42345, 0.437466, 0.446473, 0.46018, 0.46899,
          0.4824, 0.491022, 0.504149, 0.51259, 0.525444, 0.533712, 0.546306,
          0.554408, 0.56675}




          (takes about 5 seconds)



          Alternatively, use



          Table[u[i], {i, 1, 100}]


          (same result). Your combination of For and Print shows the results but doesn't let you keep using them for more calculations.






          share|improve this answer











          $endgroup$













          • $begingroup$
            thank you very much. I really appreciate it.
            $endgroup$
            – morapi
            5 mins ago














          5












          5








          5





          $begingroup$

          I recommend you learn the distinction between immediate (=) and delayed (:=) assignments. They make the difference between slow and fast code here. Start with this tutorial or this book chapter, then look at memoization.



          s = 0.3405;
          
e = 1.6539*10^-21;
          
u[0] = 0.;
          
u[1] = 0.1;
          

          
A[r_] = Piecewise[{{r - 2.5 s - 48*e*s^12*r^-13 + 24*e*s^6*r^-7, r > 2.5 s},
          
                   {-48*e*s^12*r^-13 + 24*e*s^6*r^-7, s <= r <= 2.5 s},
          
                   {r - s - 24*e*s^-1, r < s}}];
          

          
u[i_] := u[i] = x /. FindRoot[
          
  u[i - 1] + 1/(i^2 (u[i - 1] - u[i - 2])^2) (u[i - 1] - u[i - 2]) - 0.9 A[x] == x, {x, 1.}]
          

          
Array[u, 100]



          {0.1, 1.77164, 1.37065, 1.04259, 0.887781, 0.708344, 0.59461,
          0.457228, 0.367364, 0.296071, 0.256104, 0.20463, 0.208487, 1.20917,
          1.04197, 0.939331, 0.879865, 0.827963, 0.774591, 0.72775, 0.67934,
          0.63666, 0.592369, 0.553172, 0.512352, 0.476112, 0.438261, 0.404563,
          0.369277, 0.339073, 0.321616, 0.301118, 0.296195, 0.224688, 0.273538,
          0.31357, 0.33593, 0.366902, 0.38813, 0.417572, 0.437777, 0.465834,
          0.48511, 0.511907, 0.530336, 0.55598, 0.573633, 0.598219, 0.615159,
          0.638772, 0.655054, 0.677768, 0.693441, 0.715321, 0.73043, 0.751535,
          0.766118, 0.786503, 0.800596, 0.820306, 0.833941, 0.852182, 0.85901,
          0.874152, 0.871531, 0.78396, 0.781416, 0.696402, 0.693931, 0.611329,
          0.608927, 0.528603, 0.526267, 0.448099, 0.445825, 0.369701, 0.367485,
          0.315658, 0.325798, 0.341207, 0.351098, 0.366134, 0.375788, 0.390468,
          0.399897, 0.414237, 0.42345, 0.437466, 0.446473, 0.46018, 0.46899,
          0.4824, 0.491022, 0.504149, 0.51259, 0.525444, 0.533712, 0.546306,
          0.554408, 0.56675}




          (takes about 5 seconds)



          Alternatively, use



          Table[u[i], {i, 1, 100}]


          (same result). Your combination of For and Print shows the results but doesn't let you keep using them for more calculations.






          share|improve this answer











          $endgroup$



          I recommend you learn the distinction between immediate (=) and delayed (:=) assignments. They make the difference between slow and fast code here. Start with this tutorial or this book chapter, then look at memoization.



          s = 0.3405;
          
e = 1.6539*10^-21;
          
u[0] = 0.;
          
u[1] = 0.1;
          

          
A[r_] = Piecewise[{{r - 2.5 s - 48*e*s^12*r^-13 + 24*e*s^6*r^-7, r > 2.5 s},
          
                   {-48*e*s^12*r^-13 + 24*e*s^6*r^-7, s <= r <= 2.5 s},
          
                   {r - s - 24*e*s^-1, r < s}}];
          

          
u[i_] := u[i] = x /. FindRoot[
          
  u[i - 1] + 1/(i^2 (u[i - 1] - u[i - 2])^2) (u[i - 1] - u[i - 2]) - 0.9 A[x] == x, {x, 1.}]
          

          
Array[u, 100]



          {0.1, 1.77164, 1.37065, 1.04259, 0.887781, 0.708344, 0.59461,
          0.457228, 0.367364, 0.296071, 0.256104, 0.20463, 0.208487, 1.20917,
          1.04197, 0.939331, 0.879865, 0.827963, 0.774591, 0.72775, 0.67934,
          0.63666, 0.592369, 0.553172, 0.512352, 0.476112, 0.438261, 0.404563,
          0.369277, 0.339073, 0.321616, 0.301118, 0.296195, 0.224688, 0.273538,
          0.31357, 0.33593, 0.366902, 0.38813, 0.417572, 0.437777, 0.465834,
          0.48511, 0.511907, 0.530336, 0.55598, 0.573633, 0.598219, 0.615159,
          0.638772, 0.655054, 0.677768, 0.693441, 0.715321, 0.73043, 0.751535,
          0.766118, 0.786503, 0.800596, 0.820306, 0.833941, 0.852182, 0.85901,
          0.874152, 0.871531, 0.78396, 0.781416, 0.696402, 0.693931, 0.611329,
          0.608927, 0.528603, 0.526267, 0.448099, 0.445825, 0.369701, 0.367485,
          0.315658, 0.325798, 0.341207, 0.351098, 0.366134, 0.375788, 0.390468,
          0.399897, 0.414237, 0.42345, 0.437466, 0.446473, 0.46018, 0.46899,
          0.4824, 0.491022, 0.504149, 0.51259, 0.525444, 0.533712, 0.546306,
          0.554408, 0.56675}




          (takes about 5 seconds)



          Alternatively, use



          Table[u[i], {i, 1, 100}]


          (same result). Your combination of For and Print shows the results but doesn't let you keep using them for more calculations.







          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited 1 hour ago

























          answered 1 hour ago









          RomanRoman

          5,12011130




          5,12011130












          • $begingroup$
            thank you very much. I really appreciate it.
            $endgroup$
            – morapi
            5 mins ago


















          • $begingroup$
            thank you very much. I really appreciate it.
            $endgroup$
            – morapi
            5 mins ago
















          $begingroup$
          thank you very much. I really appreciate it.
          $endgroup$
          – morapi
          5 mins ago




          $begingroup$
          thank you very much. I really appreciate it.
          $endgroup$
          – morapi
          5 mins ago


















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          Старые Смолеговицы Содержание История | География | Демография | Достопримечательности | Примечания | НавигацияHGЯOLHGЯOL41 206 832 01641 606 406 141Административно-территориальное деление Ленинградской области«Переписная оброчная книга Водской пятины 1500 года», С. 793«Карта Ингерманландии: Ивангорода, Яма, Копорья, Нотеборга», по материалам 1676 г.«Генеральная карта провинции Ингерманландии» Э. Белинга и А. Андерсина, 1704 г., составлена по материалам 1678 г.«Географический чертёж над Ижорскою землей со своими городами» Адриана Шонбека 1705 г.Новая и достоверная всей Ингерманландии ланткарта. Грав. А. Ростовцев. СПб., 1727 г.Топографическая карта Санкт-Петербургской губернии. 5-и верстка. Шуберт. 1834 г.Описание Санкт-Петербургской губернии по уездам и станамСпецкарта западной части России Ф. Ф. Шуберта. 1844 г.Алфавитный список селений по уездам и станам С.-Петербургской губернииСписки населённых мест Российской Империи, составленные и издаваемые центральным статистическим комитетом министерства внутренних дел. XXXVII. Санкт-Петербургская губерния. По состоянию на 1862 год. СПб. 1864. С. 203Материалы по статистике народного хозяйства в С.-Петербургской губернии. Вып. IX. Частновладельческое хозяйство в Ямбургском уезде. СПб, 1888, С. 146, С. 2, 7, 54Положение о гербе муниципального образования Курское сельское поселениеСправочник истории административно-территориального деления Ленинградской области.Топографическая карта Ленинградской области, квадрат О-35-23-В (Хотыницы), 1930 г.АрхивированоАдминистративно-территориальное деление Ленинградской области. — Л., 1933, С. 27, 198АрхивированоАдминистративно-экономический справочник по Ленинградской области. — Л., 1936, с. 219АрхивированоАдминистративно-территориальное деление Ленинградской области. — Л., 1966, с. 175АрхивированоАдминистративно-территориальное деление Ленинградской области. — Лениздат, 1973, С. 180АрхивированоАдминистративно-территориальное деление Ленинградской области. — Лениздат, 1990, ISBN 5-289-00612-5, С. 38АрхивированоАдминистративно-территориальное деление Ленинградской области. — СПб., 2007, с. 60АрхивированоКоряков Юрий База данных «Этно-языковой состав населённых пунктов России». Ленинградская область.Административно-территориальное деление Ленинградской области. — СПб, 1997, ISBN 5-86153-055-6, С. 41АрхивированоКультовый комплекс Старые Смолеговицы // Электронная энциклопедия ЭрмитажаПроблемы выявления, изучения и сохранения культовых комплексов с каменными крестами: по материалам работ 2016-2017 гг. в Ленинградской области

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          Населённые пункты по алфавитуНаселённые пункты Волосовского районаМызы Ингрии Беседском сельском поселенииВолосовского районаЛенинградской областиПисцово...
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