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Numerical value of Determinant far from what it is supposed to be


Numeric values from transformFindRoot for numerical function and how to search more than one rootCan't get a numerical value for this vector operationNumerical value of an expression with a uniform step sizeNumerical evaluation after Normal?Numerical Results with replacement rulesStrange numerical valuesWhat is the acceptable error in numerical calculations?High numerical precision failingHow to extract the minimal value from NMinimize?













1












$begingroup$


I have a large matrix with numerical components and want to set the determinant to zero using the parameter h (see below). Naively, I would have expected that h sets the determinant to (approximately) zero, which isn't the case. On top of that, the order of applying the rule sol seems to affects the final outcome for a reason to don't see.



My output of the code below is:



h -> -0.744736 + 4.42008 I

0.0445865 - 0.0285418 I

0.0545654 - 0.114258 I


I am not familiar with how Mathematica handles floating point numbers so that's probably where my error lies. I have also tried to increase the precision with SetPrecision, but without success.



mat=0.16 - (0.36 + 0.001 I) h - (1.35808 - 
 0.00120116 I) h^2 - (0.49603 - 0.00137214 I) h^3 - (0.11307 - 
 0.00105331 I) h^4 + (0.249794 - 0.000384238 I) h^5 - 
0.39204 h^6, -0.1711 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h - (1.15528 + 
 0.00267142 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), (0.0000353051 - 1.67323*10^-6 I) h^4, 
0, -0.1711 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (19.6394 - 
 0.00267142 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (11.3534 - 
 0.00119507 I) h^2 - (0.484268 - 0.00140481 I) h^3 - (5.0714 - 
 0.00114074 I) h^4 + (0.27061 - 0.000416258 I) h^5 - 
0.42471 h^6, -0.223386 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (4.95742 - 
 0.00267502 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), (0.0000484431 - 2.29589*10^-6 I) h^4, (0.0000353051 - 
 1.67323*10^-6 I) h^4, -0.223386 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (41.4016 - 
 0.00267502 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (29.348 - 
 0.00118803 I) h^2 - (0.470698 - 0.00144251 I) h^3 - (13.9095 - 
 0.00124106 I) h^4 + (0.294629 - 0.000453204 I) h^5 - 
0.462406 h^6, -0.234771 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (19.0123 - 
 0.00267319 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), 0, (0.0000484431 - 
 2.29589*10^-6 I) h^4, -0.234771 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (71.32 - 
 0.00267319 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (55.3462 - 
 0.00118568 I) h^2 - (0.466163 - 0.0014551 I) h^3 - (26.6556 - 
 0.00127449 I) h^4 + (0.302655 - 0.000465551 I) h^5 - 
0.475003 h^6;
sol = Part[NSolve[Det[%] == 0, h], 1]
Det[mat /. sol]
Det[mat] /. sol





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  • $begingroup$
    Correction: I get the output 0.118714 - 0.0526506 I (as the second output) and 0.106201 - 0.0979004 I (as the third output); sorry, used a different matrix. But the problem still stands.
    $endgroup$
    – Nils
    2 hours ago















1












$begingroup$


I have a large matrix with numerical components and want to set the determinant to zero using the parameter h (see below). Naively, I would have expected that h sets the determinant to (approximately) zero, which isn't the case. On top of that, the order of applying the rule sol seems to affects the final outcome for a reason to don't see.



My output of the code below is:



h -> -0.744736 + 4.42008 I

0.0445865 - 0.0285418 I

0.0545654 - 0.114258 I


I am not familiar with how Mathematica handles floating point numbers so that's probably where my error lies. I have also tried to increase the precision with SetPrecision, but without success.



mat=0.16 - (0.36 + 0.001 I) h - (1.35808 - 
 0.00120116 I) h^2 - (0.49603 - 0.00137214 I) h^3 - (0.11307 - 
 0.00105331 I) h^4 + (0.249794 - 0.000384238 I) h^5 - 
0.39204 h^6, -0.1711 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h - (1.15528 + 
 0.00267142 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), (0.0000353051 - 1.67323*10^-6 I) h^4, 
0, -0.1711 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (19.6394 - 
 0.00267142 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (11.3534 - 
 0.00119507 I) h^2 - (0.484268 - 0.00140481 I) h^3 - (5.0714 - 
 0.00114074 I) h^4 + (0.27061 - 0.000416258 I) h^5 - 
0.42471 h^6, -0.223386 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (4.95742 - 
 0.00267502 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), (0.0000484431 - 2.29589*10^-6 I) h^4, (0.0000353051 - 
 1.67323*10^-6 I) h^4, -0.223386 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (41.4016 - 
 0.00267502 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (29.348 - 
 0.00118803 I) h^2 - (0.470698 - 0.00144251 I) h^3 - (13.9095 - 
 0.00124106 I) h^4 + (0.294629 - 0.000453204 I) h^5 - 
0.462406 h^6, -0.234771 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (19.0123 - 
 0.00267319 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), 0, (0.0000484431 - 
 2.29589*10^-6 I) h^4, -0.234771 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (71.32 - 
 0.00267319 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (55.3462 - 
 0.00118568 I) h^2 - (0.466163 - 0.0014551 I) h^3 - (26.6556 - 
 0.00127449 I) h^4 + (0.302655 - 0.000465551 I) h^5 - 
0.475003 h^6;
sol = Part[NSolve[Det[%] == 0, h], 1]
Det[mat /. sol]
Det[mat] /. sol





numerical-value







share|improve this question







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Nils is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$











  • $begingroup$
    Correction: I get the output 0.118714 - 0.0526506 I (as the second output) and 0.106201 - 0.0979004 I (as the third output); sorry, used a different matrix. But the problem still stands.
    $endgroup$
    – Nils
    2 hours ago













1












1








1





$begingroup$


I have a large matrix with numerical components and want to set the determinant to zero using the parameter h (see below). Naively, I would have expected that h sets the determinant to (approximately) zero, which isn't the case. On top of that, the order of applying the rule sol seems to affects the final outcome for a reason to don't see.



My output of the code below is:



h -> -0.744736 + 4.42008 I

0.0445865 - 0.0285418 I

0.0545654 - 0.114258 I


I am not familiar with how Mathematica handles floating point numbers so that's probably where my error lies. I have also tried to increase the precision with SetPrecision, but without success.



mat=0.16 - (0.36 + 0.001 I) h - (1.35808 - 
 0.00120116 I) h^2 - (0.49603 - 0.00137214 I) h^3 - (0.11307 - 
 0.00105331 I) h^4 + (0.249794 - 0.000384238 I) h^5 - 
0.39204 h^6, -0.1711 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h - (1.15528 + 
 0.00267142 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), (0.0000353051 - 1.67323*10^-6 I) h^4, 
0, -0.1711 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (19.6394 - 
 0.00267142 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (11.3534 - 
 0.00119507 I) h^2 - (0.484268 - 0.00140481 I) h^3 - (5.0714 - 
 0.00114074 I) h^4 + (0.27061 - 0.000416258 I) h^5 - 
0.42471 h^6, -0.223386 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (4.95742 - 
 0.00267502 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), (0.0000484431 - 2.29589*10^-6 I) h^4, (0.0000353051 - 
 1.67323*10^-6 I) h^4, -0.223386 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (41.4016 - 
 0.00267502 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (29.348 - 
 0.00118803 I) h^2 - (0.470698 - 0.00144251 I) h^3 - (13.9095 - 
 0.00124106 I) h^4 + (0.294629 - 0.000453204 I) h^5 - 
0.462406 h^6, -0.234771 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (19.0123 - 
 0.00267319 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), 0, (0.0000484431 - 
 2.29589*10^-6 I) h^4, -0.234771 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (71.32 - 
 0.00267319 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (55.3462 - 
 0.00118568 I) h^2 - (0.466163 - 0.0014551 I) h^3 - (26.6556 - 
 0.00127449 I) h^4 + (0.302655 - 0.000465551 I) h^5 - 
0.475003 h^6;
sol = Part[NSolve[Det[%] == 0, h], 1]
Det[mat /. sol]
Det[mat] /. sol





numerical-value







share|improve this question







New contributor




Nils 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 large matrix with numerical components and want to set the determinant to zero using the parameter h (see below). Naively, I would have expected that h sets the determinant to (approximately) zero, which isn't the case. On top of that, the order of applying the rule sol seems to affects the final outcome for a reason to don't see.



My output of the code below is:



h -> -0.744736 + 4.42008 I

0.0445865 - 0.0285418 I

0.0545654 - 0.114258 I


I am not familiar with how Mathematica handles floating point numbers so that's probably where my error lies. I have also tried to increase the precision with SetPrecision, but without success.



mat=0.16 - (0.36 + 0.001 I) h - (1.35808 - 
 0.00120116 I) h^2 - (0.49603 - 0.00137214 I) h^3 - (0.11307 - 
 0.00105331 I) h^4 + (0.249794 - 0.000384238 I) h^5 - 
0.39204 h^6, -0.1711 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h - (1.15528 + 
 0.00267142 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), (0.0000353051 - 1.67323*10^-6 I) h^4, 
0, -0.1711 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (19.6394 - 
 0.00267142 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (11.3534 - 
 0.00119507 I) h^2 - (0.484268 - 0.00140481 I) h^3 - (5.0714 - 
 0.00114074 I) h^4 + (0.27061 - 0.000416258 I) h^5 - 
0.42471 h^6, -0.223386 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (4.95742 - 
 0.00267502 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), (0.0000484431 - 2.29589*10^-6 I) h^4, (0.0000353051 - 
 1.67323*10^-6 I) h^4, -0.223386 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (41.4016 - 
 0.00267502 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (29.348 - 
 0.00118803 I) h^2 - (0.470698 - 0.00144251 I) h^3 - (13.9095 - 
 0.00124106 I) h^4 + (0.294629 - 0.000453204 I) h^5 - 
0.462406 h^6, -0.234771 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (19.0123 - 
 0.00267319 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 
 1. h^4), 0, (0.0000484431 - 
 2.29589*10^-6 I) h^4, -0.234771 h^2 ((-0.143205 + 
 0.000186623 I) - (0.36 + 0.001 I) h + (71.32 - 
 0.00267319 I) h^2 - (0.637164 - 0.0009801 I) h^3 + 1. h^4), 
0.16 - (0.36 + 0.001 I) h - (55.3462 - 
 0.00118568 I) h^2 - (0.466163 - 0.0014551 I) h^3 - (26.6556 - 
 0.00127449 I) h^4 + (0.302655 - 0.000465551 I) h^5 - 
0.475003 h^6;
sol = Part[NSolve[Det[%] == 0, h], 1]
Det[mat /. sol]
Det[mat] /. sol





numerical-value




numerical-value






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share|improve this question







New contributor




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Check out our Code of Conduct.









share|improve this question




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asked 2 hours ago









NilsNils

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New contributor





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






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











  • $begingroup$
    Correction: I get the output 0.118714 - 0.0526506 I (as the second output) and 0.106201 - 0.0979004 I (as the third output); sorry, used a different matrix. But the problem still stands.
    $endgroup$
    – Nils
    2 hours ago
















  • $begingroup$
    Correction: I get the output 0.118714 - 0.0526506 I (as the second output) and 0.106201 - 0.0979004 I (as the third output); sorry, used a different matrix. But the problem still stands.
    $endgroup$
    – Nils
    2 hours ago















$begingroup$
Correction: I get the output 0.118714 - 0.0526506 I (as the second output) and 0.106201 - 0.0979004 I (as the third output); sorry, used a different matrix. But the problem still stands.
$endgroup$
– Nils
2 hours ago




$begingroup$
Correction: I get the output 0.118714 - 0.0526506 I (as the second output) and 0.106201 - 0.0979004 I (as the third output); sorry, used a different matrix. But the problem still stands.
$endgroup$
– Nils
2 hours ago










1 Answer
1






active

oldest

votes


















3












$begingroup$

As you suspected when you mentioned SetPrecision, you are encountering numerical errors, probably catastrophic loss of precision when calculating the determinant; your calculations do in fact need to be carried out at higher precision.



If possible, you would want to use exact numbers in your matrix, or take advantage of the arbitrary-precision capabilities of Mathematica. For instance, we can convert all machine-precision numbers to arbitrary-precision ones with a number of digits of precision equal to that of common machine-precision numbers on your machine using SetPrecision (see also $MachinePrecision in the documentation):



det = Det[SetPrecision[mat, $MachinePrecision]];
sol = NSolve[det == 0, h];
det /. sol // PossibleZeroQ

(* Out: 
 True, True, True, True, True, True, True, True, True, True, True,
 True, True, True, True, True, True, True, True, True, True, True,
 True, True 
*)


As you can see, all those values of $h$ do bring your determinant reasonably close to zero, within machine-precision approximations.






share|improve this answer











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    1 Answer
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    1 Answer
    1






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    3












    $begingroup$

    As you suspected when you mentioned SetPrecision, you are encountering numerical errors, probably catastrophic loss of precision when calculating the determinant; your calculations do in fact need to be carried out at higher precision.



    If possible, you would want to use exact numbers in your matrix, or take advantage of the arbitrary-precision capabilities of Mathematica. For instance, we can convert all machine-precision numbers to arbitrary-precision ones with a number of digits of precision equal to that of common machine-precision numbers on your machine using SetPrecision (see also $MachinePrecision in the documentation):



    det = Det[SetPrecision[mat, $MachinePrecision]];
    sol = NSolve[det == 0, h];
    det /. sol // PossibleZeroQ

    (* Out: 
     True, True, True, True, True, True, True, True, True, True, True,
     True, True, True, True, True, True, True, True, True, True, True,
     True, True 
    *)


    As you can see, all those values of $h$ do bring your determinant reasonably close to zero, within machine-precision approximations.






    share|improve this answer











    $endgroup$

















      3












      $begingroup$

      As you suspected when you mentioned SetPrecision, you are encountering numerical errors, probably catastrophic loss of precision when calculating the determinant; your calculations do in fact need to be carried out at higher precision.



      If possible, you would want to use exact numbers in your matrix, or take advantage of the arbitrary-precision capabilities of Mathematica. For instance, we can convert all machine-precision numbers to arbitrary-precision ones with a number of digits of precision equal to that of common machine-precision numbers on your machine using SetPrecision (see also $MachinePrecision in the documentation):



      det = Det[SetPrecision[mat, $MachinePrecision]];
      sol = NSolve[det == 0, h];
      det /. sol // PossibleZeroQ

      (* Out: 
       True, True, True, True, True, True, True, True, True, True, True,
       True, True, True, True, True, True, True, True, True, True, True,
       True, True 
      *)


      As you can see, all those values of $h$ do bring your determinant reasonably close to zero, within machine-precision approximations.






      share|improve this answer











      $endgroup$















        3












        3








        3





        $begingroup$

        As you suspected when you mentioned SetPrecision, you are encountering numerical errors, probably catastrophic loss of precision when calculating the determinant; your calculations do in fact need to be carried out at higher precision.



        If possible, you would want to use exact numbers in your matrix, or take advantage of the arbitrary-precision capabilities of Mathematica. For instance, we can convert all machine-precision numbers to arbitrary-precision ones with a number of digits of precision equal to that of common machine-precision numbers on your machine using SetPrecision (see also $MachinePrecision in the documentation):



        det = Det[SetPrecision[mat, $MachinePrecision]];
        sol = NSolve[det == 0, h];
        det /. sol // PossibleZeroQ

        (* Out: 
         True, True, True, True, True, True, True, True, True, True, True,
         True, True, True, True, True, True, True, True, True, True, True,
         True, True 
        *)


        As you can see, all those values of $h$ do bring your determinant reasonably close to zero, within machine-precision approximations.






        share|improve this answer











        $endgroup$



        As you suspected when you mentioned SetPrecision, you are encountering numerical errors, probably catastrophic loss of precision when calculating the determinant; your calculations do in fact need to be carried out at higher precision.



        If possible, you would want to use exact numbers in your matrix, or take advantage of the arbitrary-precision capabilities of Mathematica. For instance, we can convert all machine-precision numbers to arbitrary-precision ones with a number of digits of precision equal to that of common machine-precision numbers on your machine using SetPrecision (see also $MachinePrecision in the documentation):



        det = Det[SetPrecision[mat, $MachinePrecision]];
        sol = NSolve[det == 0, h];
        det /. sol // PossibleZeroQ

        (* Out: 
         True, True, True, True, True, True, True, True, True, True, True,
         True, True, True, True, True, True, True, True, True, True, True,
         True, True 
        *)


        As you can see, all those values of $h$ do bring your determinant reasonably close to zero, within machine-precision approximations.







        share|improve this answer














        share|improve this answer



        share|improve this answer








        edited 2 hours ago

























        answered 2 hours ago









        MarcoBMarcoB

        37.3k556113




        37.3k556113




















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