How to prevent mathematica rounding extremely small numbers to zero?












4












$begingroup$


I have a function that, while the maths itself is unimportant, at certain values it results in a very large number multiplying a very small number. E.g. 10^450000 * 10^-449998. As you can see, this should output the more-sensible number 10^2. However Mathematica is rounding the small number to zero thus the whole calculation breaks down. How can I prevent this? I've played with MinNumber and MachinePrecision but neither seem to fix the issue.



Thanks in advance!



Edit: Including the equation as requested:



$exp[cfrac{omega^2}{sigma^2}] BesselK[1,cfrac{sqrt(cfrac{omega}{sigma^2})}{sqrt(cfrac{sigma^2}{omega^3})}]$



Equation breaks down for $sigma<0.01*omega$ and generally unreliable below $sigma<0.04*omega$



Edit2: And the Mathematica code!



bessktot[ω0_, σ_] := BesselK[1, Sqrt[ω0/σ^2]/Sqrt[σ^2/ω0^3]]

expcalctot[ω0_, σ_] := E^(ω0^2/σ^2);

ω0 = 2 [Pi] 10^12;

σ[BWpc_] := BWpc/100 ω0;

σt = σ[1]

bessktot[ω0, σt]

expcalctot[ω0, σt]

expcalctot[ω0, σt]*bessktot[ω0, σt]


Edit3: Thanks Bob Hanlon, (I can't comment back on your answer yet). Your answer works for certain values input to the bessel. The problem appears to be even more fundamental than I realised. It appears Mathematica can't calculate non-integer $BesselK[1,x]$ functions when $x>741$. Is there a way around this?






equation-solving numerics evaluation







share|improve this question









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  • $begingroup$
    On my machine, even with machine precision exponents, 10^45000000000000.*10^-44999999999998. outputs 1.0 * 10^2 from a fresh startup. It should only start rounding to 0 if it runs out of precision during the calculation, one of the intermediate functions does not handle arbitrary precision arithmetic (not common, but some don't), if it's explicitly told to start rounding somewhere, or if something is actually multiplied by 0. We will need a bit more code to diagnose what's actually going on, so please consider posting your function.
    $endgroup$
    – eyorble
    yesterday










  • $begingroup$
    @eyorblade 0^45000000000000. is certainly no a machine precision number: Precision[110^45000000000000.].
    $endgroup$
    – Henrik Schumacher
    yesterday










  • $begingroup$
    @HenrikSchumacher In a_^b_, the b is machine precision, is it not? The resulting value is not, obviously, because there's any precision tracking at all. I suspect Mathematica knows what it's doing here, in that with exact a the precision isn't lost very quickly at all, but the exponent isn't the source of the uncertainty in this case.
    $endgroup$
    – eyorble
    yesterday












  • $begingroup$
    If the issue is with Mathematica code, please post the Mathematica code rather than a LaTeX representation.
    $endgroup$
    – JimB
    yesterday






  • 2




    $begingroup$
    @Andrew When I approved your edit, I notice that you have two different accounts, both with the same name. I do not know how that happened, but you should be able to combine the two. Then, you will not need the approval of others to edit your own question.
    $endgroup$
    – bbgodfrey
    yesterday
















4












$begingroup$


I have a function that, while the maths itself is unimportant, at certain values it results in a very large number multiplying a very small number. E.g. 10^450000 * 10^-449998. As you can see, this should output the more-sensible number 10^2. However Mathematica is rounding the small number to zero thus the whole calculation breaks down. How can I prevent this? I've played with MinNumber and MachinePrecision but neither seem to fix the issue.



Thanks in advance!



Edit: Including the equation as requested:



$exp[cfrac{omega^2}{sigma^2}] BesselK[1,cfrac{sqrt(cfrac{omega}{sigma^2})}{sqrt(cfrac{sigma^2}{omega^3})}]$



Equation breaks down for $sigma<0.01*omega$ and generally unreliable below $sigma<0.04*omega$



Edit2: And the Mathematica code!



bessktot[ω0_, σ_] := BesselK[1, Sqrt[ω0/σ^2]/Sqrt[σ^2/ω0^3]]

expcalctot[ω0_, σ_] := E^(ω0^2/σ^2);

ω0 = 2 [Pi] 10^12;

σ[BWpc_] := BWpc/100 ω0;

σt = σ[1]

bessktot[ω0, σt]

expcalctot[ω0, σt]

expcalctot[ω0, σt]*bessktot[ω0, σt]


Edit3: Thanks Bob Hanlon, (I can't comment back on your answer yet). Your answer works for certain values input to the bessel. The problem appears to be even more fundamental than I realised. It appears Mathematica can't calculate non-integer $BesselK[1,x]$ functions when $x>741$. Is there a way around this?






equation-solving numerics evaluation







share|improve this question









New contributor




Andrew is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    On my machine, even with machine precision exponents, 10^45000000000000.*10^-44999999999998. outputs 1.0 * 10^2 from a fresh startup. It should only start rounding to 0 if it runs out of precision during the calculation, one of the intermediate functions does not handle arbitrary precision arithmetic (not common, but some don't), if it's explicitly told to start rounding somewhere, or if something is actually multiplied by 0. We will need a bit more code to diagnose what's actually going on, so please consider posting your function.
    $endgroup$
    – eyorble
    yesterday










  • $begingroup$
    @eyorblade 0^45000000000000. is certainly no a machine precision number: Precision[110^45000000000000.].
    $endgroup$
    – Henrik Schumacher
    yesterday










  • $begingroup$
    @HenrikSchumacher In a_^b_, the b is machine precision, is it not? The resulting value is not, obviously, because there's any precision tracking at all. I suspect Mathematica knows what it's doing here, in that with exact a the precision isn't lost very quickly at all, but the exponent isn't the source of the uncertainty in this case.
    $endgroup$
    – eyorble
    yesterday












  • $begingroup$
    If the issue is with Mathematica code, please post the Mathematica code rather than a LaTeX representation.
    $endgroup$
    – JimB
    yesterday






  • 2




    $begingroup$
    @Andrew When I approved your edit, I notice that you have two different accounts, both with the same name. I do not know how that happened, but you should be able to combine the two. Then, you will not need the approval of others to edit your own question.
    $endgroup$
    – bbgodfrey
    yesterday














4












4








4


1



$begingroup$


I have a function that, while the maths itself is unimportant, at certain values it results in a very large number multiplying a very small number. E.g. 10^450000 * 10^-449998. As you can see, this should output the more-sensible number 10^2. However Mathematica is rounding the small number to zero thus the whole calculation breaks down. How can I prevent this? I've played with MinNumber and MachinePrecision but neither seem to fix the issue.



Thanks in advance!



Edit: Including the equation as requested:



$exp[cfrac{omega^2}{sigma^2}] BesselK[1,cfrac{sqrt(cfrac{omega}{sigma^2})}{sqrt(cfrac{sigma^2}{omega^3})}]$



Equation breaks down for $sigma<0.01*omega$ and generally unreliable below $sigma<0.04*omega$



Edit2: And the Mathematica code!



bessktot[ω0_, σ_] := BesselK[1, Sqrt[ω0/σ^2]/Sqrt[σ^2/ω0^3]]

expcalctot[ω0_, σ_] := E^(ω0^2/σ^2);

ω0 = 2 [Pi] 10^12;

σ[BWpc_] := BWpc/100 ω0;

σt = σ[1]

bessktot[ω0, σt]

expcalctot[ω0, σt]

expcalctot[ω0, σt]*bessktot[ω0, σt]


Edit3: Thanks Bob Hanlon, (I can't comment back on your answer yet). Your answer works for certain values input to the bessel. The problem appears to be even more fundamental than I realised. It appears Mathematica can't calculate non-integer $BesselK[1,x]$ functions when $x>741$. Is there a way around this?






equation-solving numerics evaluation







share|improve this question









New contributor




Andrew is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




I have a function that, while the maths itself is unimportant, at certain values it results in a very large number multiplying a very small number. E.g. 10^450000 * 10^-449998. As you can see, this should output the more-sensible number 10^2. However Mathematica is rounding the small number to zero thus the whole calculation breaks down. How can I prevent this? I've played with MinNumber and MachinePrecision but neither seem to fix the issue.



Thanks in advance!



Edit: Including the equation as requested:



$exp[cfrac{omega^2}{sigma^2}] BesselK[1,cfrac{sqrt(cfrac{omega}{sigma^2})}{sqrt(cfrac{sigma^2}{omega^3})}]$



Equation breaks down for $sigma<0.01*omega$ and generally unreliable below $sigma<0.04*omega$



Edit2: And the Mathematica code!



bessktot[ω0_, σ_] := BesselK[1, Sqrt[ω0/σ^2]/Sqrt[σ^2/ω0^3]]

expcalctot[ω0_, σ_] := E^(ω0^2/σ^2);

ω0 = 2 [Pi] 10^12;

σ[BWpc_] := BWpc/100 ω0;

σt = σ[1]

bessktot[ω0, σt]

expcalctot[ω0, σt]

expcalctot[ω0, σt]*bessktot[ω0, σt]


Edit3: Thanks Bob Hanlon, (I can't comment back on your answer yet). Your answer works for certain values input to the bessel. The problem appears to be even more fundamental than I realised. It appears Mathematica can't calculate non-integer $BesselK[1,x]$ functions when $x>741$. Is there a way around this?






equation-solving numerics evaluation




equation-solving numerics evaluation






share|improve this question









New contributor




Andrew is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|improve this question









New contributor




Andrew is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|improve this question




share|improve this question








edited yesterday









Andrew

32




32






New contributor




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Check out our Code of Conduct.









asked yesterday









AndrewAndrew

212




212




New contributor




Andrew is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor





Andrew is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Andrew is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • $begingroup$
    On my machine, even with machine precision exponents, 10^45000000000000.*10^-44999999999998. outputs 1.0 * 10^2 from a fresh startup. It should only start rounding to 0 if it runs out of precision during the calculation, one of the intermediate functions does not handle arbitrary precision arithmetic (not common, but some don't), if it's explicitly told to start rounding somewhere, or if something is actually multiplied by 0. We will need a bit more code to diagnose what's actually going on, so please consider posting your function.
    $endgroup$
    – eyorble
    yesterday










  • $begingroup$
    @eyorblade 0^45000000000000. is certainly no a machine precision number: Precision[110^45000000000000.].
    $endgroup$
    – Henrik Schumacher
    yesterday










  • $begingroup$
    @HenrikSchumacher In a_^b_, the b is machine precision, is it not? The resulting value is not, obviously, because there's any precision tracking at all. I suspect Mathematica knows what it's doing here, in that with exact a the precision isn't lost very quickly at all, but the exponent isn't the source of the uncertainty in this case.
    $endgroup$
    – eyorble
    yesterday












  • $begingroup$
    If the issue is with Mathematica code, please post the Mathematica code rather than a LaTeX representation.
    $endgroup$
    – JimB
    yesterday






  • 2




    $begingroup$
    @Andrew When I approved your edit, I notice that you have two different accounts, both with the same name. I do not know how that happened, but you should be able to combine the two. Then, you will not need the approval of others to edit your own question.
    $endgroup$
    – bbgodfrey
    yesterday


















  • $begingroup$
    On my machine, even with machine precision exponents, 10^45000000000000.*10^-44999999999998. outputs 1.0 * 10^2 from a fresh startup. It should only start rounding to 0 if it runs out of precision during the calculation, one of the intermediate functions does not handle arbitrary precision arithmetic (not common, but some don't), if it's explicitly told to start rounding somewhere, or if something is actually multiplied by 0. We will need a bit more code to diagnose what's actually going on, so please consider posting your function.
    $endgroup$
    – eyorble
    yesterday










  • $begingroup$
    @eyorblade 0^45000000000000. is certainly no a machine precision number: Precision[110^45000000000000.].
    $endgroup$
    – Henrik Schumacher
    yesterday










  • $begingroup$
    @HenrikSchumacher In a_^b_, the b is machine precision, is it not? The resulting value is not, obviously, because there's any precision tracking at all. I suspect Mathematica knows what it's doing here, in that with exact a the precision isn't lost very quickly at all, but the exponent isn't the source of the uncertainty in this case.
    $endgroup$
    – eyorble
    yesterday












  • $begingroup$
    If the issue is with Mathematica code, please post the Mathematica code rather than a LaTeX representation.
    $endgroup$
    – JimB
    yesterday






  • 2




    $begingroup$
    @Andrew When I approved your edit, I notice that you have two different accounts, both with the same name. I do not know how that happened, but you should be able to combine the two. Then, you will not need the approval of others to edit your own question.
    $endgroup$
    – bbgodfrey
    yesterday
















$begingroup$
On my machine, even with machine precision exponents, 10^45000000000000.*10^-44999999999998. outputs 1.0 * 10^2 from a fresh startup. It should only start rounding to 0 if it runs out of precision during the calculation, one of the intermediate functions does not handle arbitrary precision arithmetic (not common, but some don't), if it's explicitly told to start rounding somewhere, or if something is actually multiplied by 0. We will need a bit more code to diagnose what's actually going on, so please consider posting your function.
$endgroup$
– eyorble
yesterday




$begingroup$
On my machine, even with machine precision exponents, 10^45000000000000.*10^-44999999999998. outputs 1.0 * 10^2 from a fresh startup. It should only start rounding to 0 if it runs out of precision during the calculation, one of the intermediate functions does not handle arbitrary precision arithmetic (not common, but some don't), if it's explicitly told to start rounding somewhere, or if something is actually multiplied by 0. We will need a bit more code to diagnose what's actually going on, so please consider posting your function.
$endgroup$
– eyorble
yesterday












$begingroup$
@eyorblade 0^45000000000000. is certainly no a machine precision number: Precision[110^45000000000000.].
$endgroup$
– Henrik Schumacher
yesterday




$begingroup$
@eyorblade 0^45000000000000. is certainly no a machine precision number: Precision[110^45000000000000.].
$endgroup$
– Henrik Schumacher
yesterday












$begingroup$
@HenrikSchumacher In a_^b_, the b is machine precision, is it not? The resulting value is not, obviously, because there's any precision tracking at all. I suspect Mathematica knows what it's doing here, in that with exact a the precision isn't lost very quickly at all, but the exponent isn't the source of the uncertainty in this case.
$endgroup$
– eyorble
yesterday






$begingroup$
@HenrikSchumacher In a_^b_, the b is machine precision, is it not? The resulting value is not, obviously, because there's any precision tracking at all. I suspect Mathematica knows what it's doing here, in that with exact a the precision isn't lost very quickly at all, but the exponent isn't the source of the uncertainty in this case.
$endgroup$
– eyorble
yesterday














$begingroup$
If the issue is with Mathematica code, please post the Mathematica code rather than a LaTeX representation.
$endgroup$
– JimB
yesterday




$begingroup$
If the issue is with Mathematica code, please post the Mathematica code rather than a LaTeX representation.
$endgroup$
– JimB
yesterday




2




2




$begingroup$
@Andrew When I approved your edit, I notice that you have two different accounts, both with the same name. I do not know how that happened, but you should be able to combine the two. Then, you will not need the approval of others to edit your own question.
$endgroup$
– bbgodfrey
yesterday




$begingroup$
@Andrew When I approved your edit, I notice that you have two different accounts, both with the same name. I do not know how that happened, but you should be able to combine the two. Then, you will not need the approval of others to edit your own question.
$endgroup$
– bbgodfrey
yesterday










1 Answer
1






active

oldest

votes


















7












$begingroup$

There was a change in handling underflow in later versions.



$Version



(* "11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018)" *)



Clear["Global`*"]



bessktot[ω0_, σ_] := 

  BesselK[1, Sqrt[ω0/σ^2]/Sqrt[σ^2/ω0^3]];

expcalctot[ω0_, σ_] := E^(ω0^2/σ^2);

ω0 = 2 π 10^12;

σ[BWpc_] := BWpc/100 ω0;

σt = σ[1];


MachinePrecision is insufficient, 



(expr = expcalctot[ω0, σt]*bessktot[ω0, σt]) // N



(* 0. *)


Use arbitrary precision



expr // N[#, $MachinePrecision] &



(* 0.01253361135127051 *)



expr // N[#, 20] &



(* 0.012533611351270505734 *)


EDIT: For large input to BesselK you need to control the precision of the input.



BesselK[1, #] & /@ {801., 801.`20, SetPrecision[801., 20]}



(* {0., 5.9781508629496523*10^-350, 5.9781508629496523*10^-350} *)





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    1 Answer
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    active

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    oldest

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    active

    oldest

    votes









    7












    $begingroup$

    There was a change in handling underflow in later versions.



    $Version
    

    
(* "11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018)" *)
    

    
Clear["Global`*"]
    

    
bessktot[ω0_, σ_] := 
    
  BesselK[1, Sqrt[ω0/σ^2]/Sqrt[σ^2/ω0^3]];
    
expcalctot[ω0_, σ_] := E^(ω0^2/σ^2);
    
ω0 = 2 π 10^12;
    
σ[BWpc_] := BWpc/100 ω0;
    
σt = σ[1];


    MachinePrecision is insufficient, 



    (expr = expcalctot[ω0, σt]*bessktot[ω0, σt]) // N
    

    
(* 0. *)


    Use arbitrary precision



    expr // N[#, $MachinePrecision] &
    

    
(* 0.01253361135127051 *)
    

    
expr // N[#, 20] &
    

    
(* 0.012533611351270505734 *)


    EDIT: For large input to BesselK you need to control the precision of the input.



    BesselK[1, #] & /@ {801., 801.`20, SetPrecision[801., 20]}
    

    
(* {0., 5.9781508629496523*10^-350, 5.9781508629496523*10^-350} *)





    share|improve this answer











    $endgroup$


















      7












      $begingroup$

      There was a change in handling underflow in later versions.



      $Version
      

      
(* "11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018)" *)
      

      
Clear["Global`*"]
      

      
bessktot[ω0_, σ_] := 
      
  BesselK[1, Sqrt[ω0/σ^2]/Sqrt[σ^2/ω0^3]];
      
expcalctot[ω0_, σ_] := E^(ω0^2/σ^2);
      
ω0 = 2 π 10^12;
      
σ[BWpc_] := BWpc/100 ω0;
      
σt = σ[1];


      MachinePrecision is insufficient, 



      (expr = expcalctot[ω0, σt]*bessktot[ω0, σt]) // N
      

      
(* 0. *)


      Use arbitrary precision



      expr // N[#, $MachinePrecision] &
      

      
(* 0.01253361135127051 *)
      

      
expr // N[#, 20] &
      

      
(* 0.012533611351270505734 *)


      EDIT: For large input to BesselK you need to control the precision of the input.



      BesselK[1, #] & /@ {801., 801.`20, SetPrecision[801., 20]}
      

      
(* {0., 5.9781508629496523*10^-350, 5.9781508629496523*10^-350} *)





      share|improve this answer











      $endgroup$
















        7












        7








        7





        $begingroup$

        There was a change in handling underflow in later versions.



        $Version
        

        
(* "11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018)" *)
        

        
Clear["Global`*"]
        

        
bessktot[ω0_, σ_] := 
        
  BesselK[1, Sqrt[ω0/σ^2]/Sqrt[σ^2/ω0^3]];
        
expcalctot[ω0_, σ_] := E^(ω0^2/σ^2);
        
ω0 = 2 π 10^12;
        
σ[BWpc_] := BWpc/100 ω0;
        
σt = σ[1];


        MachinePrecision is insufficient, 



        (expr = expcalctot[ω0, σt]*bessktot[ω0, σt]) // N
        

        
(* 0. *)


        Use arbitrary precision



        expr // N[#, $MachinePrecision] &
        

        
(* 0.01253361135127051 *)
        

        
expr // N[#, 20] &
        

        
(* 0.012533611351270505734 *)


        EDIT: For large input to BesselK you need to control the precision of the input.



        BesselK[1, #] & /@ {801., 801.`20, SetPrecision[801., 20]}
        

        
(* {0., 5.9781508629496523*10^-350, 5.9781508629496523*10^-350} *)





        share|improve this answer











        $endgroup$



        There was a change in handling underflow in later versions.



        $Version
        

        
(* "11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018)" *)
        

        
Clear["Global`*"]
        

        
bessktot[ω0_, σ_] := 
        
  BesselK[1, Sqrt[ω0/σ^2]/Sqrt[σ^2/ω0^3]];
        
expcalctot[ω0_, σ_] := E^(ω0^2/σ^2);
        
ω0 = 2 π 10^12;
        
σ[BWpc_] := BWpc/100 ω0;
        
σt = σ[1];


        MachinePrecision is insufficient, 



        (expr = expcalctot[ω0, σt]*bessktot[ω0, σt]) // N
        

        
(* 0. *)


        Use arbitrary precision



        expr // N[#, $MachinePrecision] &
        

        
(* 0.01253361135127051 *)
        

        
expr // N[#, 20] &
        

        
(* 0.012533611351270505734 *)


        EDIT: For large input to BesselK you need to control the precision of the input.



        BesselK[1, #] & /@ {801., 801.`20, SetPrecision[801., 20]}
        

        
(* {0., 5.9781508629496523*10^-350, 5.9781508629496523*10^-350} *)






        share|improve this answer














        share|improve this answer



        share|improve this answer








        edited yesterday

























        answered yesterday









        Bob HanlonBob Hanlon

        61.7k33598




        61.7k33598






















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