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corrected back quoting for NumericalMath NSequenceLimit
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Chris Chiasson
Chris Chiasson
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SequenceLimit became NumericalMath``NSequenceLimitNumericalMath`NSequenceLimit. In the past, that change broke some of the examples on NLimit that used SequenceLimit. I reported the breakage, and the examples were fixed to use the same syntax but not actually call the now-non-existent function. The only remaining reference in the documentation that I can find to the use of SequenceLimit as if it were still a function is on the last bullet point of NLimit's detailed description for its Method option: "uses SequenceLimit on constructed sequence".

Recently, I started investigating the behavior of NumericalMath``NSequenceLimitNumericalMath`NSequenceLimit because I need it in my engineering work. I feel NumericalMath``NSequenceLimitNumericalMath`NSequenceLimit has some quite confusing shortcomings in regard to the approximation it returns vs. the number of terms supplied, so I wrote my own version below from some references. I also posted a C++ gist version on GitHub.

A comparison appears below showing the convergence of partial sums of series approximations to Pi and E. Afterward, I use symbolic lists to document several parts of NumericalMath``NSequenceLimitNumericalMath`NSequenceLimit that seem... possibly wrong... to me.

Below, I illustrate the difference between the behavior of the built-in NumericalMath``NSequenceLimitNumericalMath`NSequenceLimit function vs. my sequenceLimit function for symbolic lists, like {a,b,c,d}.

SequenceLimit became NumericalMath``NSequenceLimit. In the past, that change broke some of the examples on NLimit that used SequenceLimit. I reported the breakage, and the examples were fixed to use the same syntax but not actually call the now-non-existent function. The only remaining reference in the documentation that I can find to the use of SequenceLimit as if it were still a function is on the last bullet point of NLimit's detailed description for its Method option: "uses SequenceLimit on constructed sequence".

Recently, I started investigating the behavior of NumericalMath``NSequenceLimit because I need it in my engineering work. I feel NumericalMath``NSequenceLimit has some quite confusing shortcomings in regard to the approximation it returns vs. the number of terms supplied, so I wrote my own version below from some references. I also posted a C++ gist version on GitHub.

A comparison appears below showing the convergence of partial sums of series approximations to Pi and E. Afterward, I use symbolic lists to document several parts of NumericalMath``NSequenceLimit that seem... possibly wrong... to me.

Below, I illustrate the difference between the behavior of the built-in NumericalMath``NSequenceLimit function vs. my sequenceLimit function for symbolic lists, like {a,b,c,d}.

SequenceLimit became NumericalMath`NSequenceLimit. In the past, that change broke some of the examples on NLimit that used SequenceLimit. I reported the breakage, and the examples were fixed to use the same syntax but not actually call the now-non-existent function. The only remaining reference in the documentation that I can find to the use of SequenceLimit as if it were still a function is on the last bullet point of NLimit's detailed description for its Method option: "uses SequenceLimit on constructed sequence".

Recently, I started investigating the behavior of NumericalMath`NSequenceLimit because I need it in my engineering work. I feel NumericalMath`NSequenceLimit has some quite confusing shortcomings in regard to the approximation it returns vs. the number of terms supplied, so I wrote my own version below from some references. I also posted a C++ gist version on GitHub.

A comparison appears below showing the convergence of partial sums of series approximations to Pi and E. Afterward, I use symbolic lists to document several parts of NumericalMath`NSequenceLimit that seem... possibly wrong... to me.

Below, I illustrate the difference between the behavior of the built-in NumericalMath`NSequenceLimit function vs. my sequenceLimit function for symbolic lists, like {a,b,c,d}.

commenting on lack of evaluation
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Chris Chiasson
Chris Chiasson
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Quiet@NumericalMath`NSequenceLimit[{a,b,c}]
(*NumericalMath`NSequenceLimit[{a,b,c}]*)
(*yes, unevaluated with 3 inputs*)

sequenceLimit[{a,b,c}]//Simplify (*my function*)
(*(-b^2+a*c)/(a-2*b+c)*)
(*same as Mathematica's next result with 4 inputs*)

NumericalMath`NSequenceLimit[{a,b,c,d}]
(*(-b^2+a*c)/(a-2*b+c)*)
(*yes, only references a, b, & c not the later, more converged, d*)

sequenceLimit[{a,b,c,d}]//Simplify (*my function*)
(*(-c^2+b*d)/(b-2*c+d)*)
(*note use of d, which explains why my approximations
  for Pi and E above are better*)

NumericalMath`NSequenceLimit[{a,b,c,d,e}]
(*(-b^2+a*c)/(a-2*b+c)*) 
(*yes, same result as with 4 terms, no d or e in output*)

sequenceLimit[{a,b,c,d,e}]//Simplify (*my function*)
(*(c^3+a*d^2+b^2*e-c*(2*b*d+a*e))/
  (b^2+3*c^2-2*c*d+d^2-2*b*(c+d-e)-c*e-a*(c-2*d+e))*)
(*note use of d and e*)

NumericalMath`NSequenceLimit[{a}]
(*NumericalMath`NSequenceLimit::seqw Sequence of length 1
  is too short for use with Degree -> 1*)
(*NumericalMath`NSequenceLimit::bdmtd WynnEpsilon is not
  a valid specification of a sequence limit extrapolation
  algorithm.*)
(*NumericalMath`NSequenceLimit[{a}]*)

NumericalMath`NSequenceLimit[{a},Degree->1]; (*Nulled output*)
(*NumericalMath`NSequenceLimit:optx Unknown option ° in 
  NumericalMath`NSequenceLimit[{a}]*)

NumericalMath`NSequenceLimit[{a},"Degree"->1]; (*Note quotes*)
(*NumericalMath`NSequenceLimit::optx Unknown option "Degree"
  in NumericalMath`NSequenceLimit[{a}]*)

sequenceLimit[{a}] (*sequence limit is my function*)
(*a*)

Quiet@NumericalMath`NSequenceLimit[{a,b}] (*Note Quiet & no evaluation*)
(*NumericalMath`NSequenceLimit[{a,b}]*)

sequenceLimit[{a,b}] (*my function*)
(*b*)

$Version
11(*11.3.0 for Linux x86 (64-bit) (March 7, 2018)*)

Quiet@NumericalMath`NSequenceLimit[{a,b,c}]
(*NumericalMath`NSequenceLimit[{a,b,c}]*)
(*yes, unevaluated with 3 inputs*)

sequenceLimit[{a,b,c}]//Simplify (*my function*)
(*(-b^2+a*c)/(a-2*b+c)*)
(*same as Mathematica's next result with 4 inputs*)

NumericalMath`NSequenceLimit[{a,b,c,d}]
(*(-b^2+a*c)/(a-2*b+c)*)
(*yes, only references a, b, & c not the later, more converged, d*)

sequenceLimit[{a,b,c,d}]//Simplify (*my function*)
(-c^2+b*d)/(b-2*c+d)
(*note use of d, which explains why my approximations
  for Pi and E above are better*)

NumericalMath`NSequenceLimit[{a,b,c,d,e}]
(*(-b^2+a*c)/(a-2*b+c)*) 
(*yes, same result as with 4 terms, no d or e in output*)

sequenceLimit[{a,b,c,d,e}]//Simplify (*my function*)
(*(c^3+a*d^2+b^2*e-c*(2*b*d+a*e))/
  (b^2+3*c^2-2*c*d+d^2-2*b*(c+d-e)-c*e-a*(c-2*d+e))*)
(*note use of d and e*)

NumericalMath`NSequenceLimit[{a}]
(*NumericalMath`NSequenceLimit::seqw Sequence of length 1
  is too short for use with Degree -> 1*)
(*NumericalMath`NSequenceLimit::bdmtd WynnEpsilon is not
  a valid specification of a sequence limit extrapolation
  algorithm.*)
(*NumericalMath`NSequenceLimit[{a}]*)

NumericalMath`NSequenceLimit[{a},Degree->1]; (*Nulled output*)
(*NumericalMath`NSequenceLimit:optx Unknown option ° in 
  NumericalMath`NSequenceLimit[{a}]*)

NumericalMath`NSequenceLimit[{a},"Degree"->1]; (*Note quotes*)
(*NumericalMath`NSequenceLimit::optx Unknown option "Degree"
  in NumericalMath`NSequenceLimit[{a}]*)

sequenceLimit[{a}] (*sequence limit is my function*)
(*a*)

Quiet@NumericalMath`NSequenceLimit[{a,b}]
(*NumericalMath`NSequenceLimit[{a,b}]*)

sequenceLimit[{a,b}] (*my function*)
(*b*)

$Version
11.3.0 for Linux x86 (64-bit) (March 7, 2018)

Quiet@NumericalMath`NSequenceLimit[{a,b,c}]
(*NumericalMath`NSequenceLimit[{a,b,c}]*)
(*yes, unevaluated with 3 inputs*)

sequenceLimit[{a,b,c}]//Simplify (*my function*)
(*(-b^2+a*c)/(a-2*b+c)*)
(*same as Mathematica's next result with 4 inputs*)

NumericalMath`NSequenceLimit[{a,b,c,d}]
(*(-b^2+a*c)/(a-2*b+c)*)
(*yes, only references a, b, & c not the later, more converged, d*)

sequenceLimit[{a,b,c,d}]//Simplify (*my function*)
(*(-c^2+b*d)/(b-2*c+d)*)
(*note use of d, which explains why my approximations
  for Pi and E above are better*)

NumericalMath`NSequenceLimit[{a,b,c,d,e}]
(*(-b^2+a*c)/(a-2*b+c)*) 
(*yes, same result as with 4 terms, no d or e in output*)

sequenceLimit[{a,b,c,d,e}]//Simplify (*my function*)
(*(c^3+a*d^2+b^2*e-c*(2*b*d+a*e))/
  (b^2+3*c^2-2*c*d+d^2-2*b*(c+d-e)-c*e-a*(c-2*d+e))*)
(*note use of d and e*)

NumericalMath`NSequenceLimit[{a}]
(*NumericalMath`NSequenceLimit::seqw Sequence of length 1
  is too short for use with Degree -> 1*)
(*NumericalMath`NSequenceLimit::bdmtd WynnEpsilon is not
  a valid specification of a sequence limit extrapolation
  algorithm.*)
(*NumericalMath`NSequenceLimit[{a}]*)

NumericalMath`NSequenceLimit[{a},Degree->1]; (*Nulled output*)
(*NumericalMath`NSequenceLimit:optx Unknown option ° in 
  NumericalMath`NSequenceLimit[{a}]*)

NumericalMath`NSequenceLimit[{a},"Degree"->1]; (*Note quotes*)
(*NumericalMath`NSequenceLimit::optx Unknown option "Degree"
  in NumericalMath`NSequenceLimit[{a}]*)

sequenceLimit[{a}] (*sequence limit is my function*)
(*a*)

Quiet@NumericalMath`NSequenceLimit[{a,b}] (*Note Quiet & no evaluation*)
(*NumericalMath`NSequenceLimit[{a,b}]*)

sequenceLimit[{a,b}] (*my function*)
(*b*)

$Version
(*11.3.0 for Linux x86 (64-bit) (March 7, 2018)*)
remove Quiet and Simplify from one of the built-in function calls
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Chris Chiasson
Chris Chiasson
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Quiet@NumericalMath`NSequenceLimit[{a,b,c}]
(*NumericalMath`NSequenceLimit[{a,b,c}]*)
(*yes, unevaluated with 3 inputs*)

sequenceLimit[{a,b,c}]//Simplify (*my function*)
(*(-b^2+a*c)/(a-2*b+c)*)
(*same as Mathematica's next result with 4 inputs*)

NumericalMath`NSequenceLimit[{a,b,c,d}]
(*(-b^2+a*c)/(a-2*b+c)*)
(*yes, only references a, b, & c not the later, more converged, d*)

sequenceLimit[{a,b,c,d}]//Simplify (*my function*)
(-c^2+b*d)/(b-2*c+d)
(*note use of d, which explains why my approximations
  for Pi and E above are better*)

Quiet@NumericalMath`NSequenceLimit[NumericalMath`NSequenceLimit[{a,b,c,d,e}]//Simplify
(*(-b^2+a*c)/(a-2*b+c)*) 
(*yes, same result as with 4 terms, no d or e in output*)

sequenceLimit[{a,b,c,d,e}]//Simplify (*my function*)
(*(c^3+a*d^2+b^2*e-c*(2*b*d+a*e))/
  (b^2+3*c^2-2*c*d+d^2-2*b*(c+d-e)-c*e-a*(c-2*d+e))*)
(*note use of d and e*)

Quiet@NumericalMath`NSequenceLimit[{a,b,c}]
(*NumericalMath`NSequenceLimit[{a,b,c}]*)
(*yes, unevaluated with 3 inputs*)

sequenceLimit[{a,b,c}]//Simplify (*my function*)
(*(-b^2+a*c)/(a-2*b+c)*)
(*same as Mathematica's next result with 4 inputs*)

NumericalMath`NSequenceLimit[{a,b,c,d}]
(*(-b^2+a*c)/(a-2*b+c)*)
(*yes, only references a, b, & c not the later, more converged, d*)

sequenceLimit[{a,b,c,d}]//Simplify (*my function*)
(-c^2+b*d)/(b-2*c+d)
(*note use of d, which explains why my approximations
  for Pi and E above are better*)

Quiet@NumericalMath`NSequenceLimit[{a,b,c,d,e}]//Simplify
(*(-b^2+a*c)/(a-2*b+c)*) 
(*yes, same result as with 4 terms, no d or e in output*)

sequenceLimit[{a,b,c,d,e}]//Simplify (*my function*)
(*(c^3+a*d^2+b^2*e-c*(2*b*d+a*e))/
  (b^2+3*c^2-2*c*d+d^2-2*b*(c+d-e)-c*e-a*(c-2*d+e))*)
(*note use of d and e*)

Quiet@NumericalMath`NSequenceLimit[{a,b,c}]
(*NumericalMath`NSequenceLimit[{a,b,c}]*)
(*yes, unevaluated with 3 inputs*)

sequenceLimit[{a,b,c}]//Simplify (*my function*)
(*(-b^2+a*c)/(a-2*b+c)*)
(*same as Mathematica's next result with 4 inputs*)

NumericalMath`NSequenceLimit[{a,b,c,d}]
(*(-b^2+a*c)/(a-2*b+c)*)
(*yes, only references a, b, & c not the later, more converged, d*)

sequenceLimit[{a,b,c,d}]//Simplify (*my function*)
(-c^2+b*d)/(b-2*c+d)
(*note use of d, which explains why my approximations
  for Pi and E above are better*)

NumericalMath`NSequenceLimit[{a,b,c,d,e}]
(*(-b^2+a*c)/(a-2*b+c)*) 
(*yes, same result as with 4 terms, no d or e in output*)

sequenceLimit[{a,b,c,d,e}]//Simplify (*my function*)
(*(c^3+a*d^2+b^2*e-c*(2*b*d+a*e))/
  (b^2+3*c^2-2*c*d+d^2-2*b*(c+d-e)-c*e-a*(c-2*d+e))*)
(*note use of d and e*)
fixed wording to make it clear the mathematica version is included in this post
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Chris Chiasson
Chris Chiasson
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shorten function more
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Chris Chiasson
Chris Chiasson
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added link to NLimit documentation
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Chris Chiasson
Chris Chiasson
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further simplification, caching
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Chris Chiasson
Chris Chiasson
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simplified my function definition
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Chris Chiasson
Chris Chiasson
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add references, and explain why my results are better
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Chris Chiasson
Chris Chiasson
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Source Link
Chris Chiasson
Chris Chiasson
  • 2k
  • 12
  • 23