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Is there a faster way to calculate Abs[z]^2 numerically?
What are some useful, undocumented Mathematica functions?Is Abs[z]^2 a bad way to calculate the square modulus of z?Replacement of Do loopsIs there a faster way to create a matrix of indices from ragged data?Is Abs[z]^2 a bad way to calculate the square modulus of z?Faster way to perform SameQ[Reduce[…], Reduce[…]]Is there a faster way to Map an Association?Faster way to extract partial data from AdjacencyMatrixIs there a package that can calculate the Ricci tensor from a numerically given metric?Is there a good way to check, whether a small value produced numerically is a symbolic zero?Any faster way to compute this?faster way to merge dataIs there a better way to calculate the semivariance of a list?
$begingroup$
Here I'm not interested in accuracy (see 13614) but rather in raw speed. You'd think that for a complex machine-precision number z, calculating Abs[z]^2 should be faster than calculating Abs[z] because the latter requires a square root whereas the former does not. Yet it's not so:
s = RandomVariate[NormalDistribution[], 10^7, 2].1, I;
Developer`PackedArrayQ[s]
(* True *)
Abs[s]^2; // AbsoluteTiming // First
(* 0.083337 *)
Abs[s]; // AbsoluteTiming // First
(* 0.033179 *)
This indicates that Abs[z]^2 is really calculated by summing the squares of real and imaginary parts, taking a square root (for Abs[z]), and then re-squaring (for Abs[z]^2).
Is there a faster way to compute Abs[z]^2? Is there a hidden equivalent to the GSL's gsl_complex_abs2 function? The source code of this GSL function is simply to return Re[z]^2+Im[z]^2; no fancy tricks.
performance-tuning numerics
asked May 10 at 14:17
RomanRoman
8,21811238
$endgroup$
add a comment |
$begingroup$
Here I'm not interested in accuracy (see 13614) but rather in raw speed. You'd think that for a complex machine-precision number z, calculating Abs[z]^2 should be faster than calculating Abs[z] because the latter requires a square root whereas the former does not. Yet it's not so:
s = RandomVariate[NormalDistribution[], 10^7, 2].1, I;
Developer`PackedArrayQ[s]
(* True *)
Abs[s]^2; // AbsoluteTiming // First
(* 0.083337 *)
Abs[s]; // AbsoluteTiming // First
(* 0.033179 *)
This indicates that Abs[z]^2 is really calculated by summing the squares of real and imaginary parts, taking a square root (for Abs[z]), and then re-squaring (for Abs[z]^2).
Is there a faster way to compute Abs[z]^2? Is there a hidden equivalent to the GSL's gsl_complex_abs2 function? The source code of this GSL function is simply to return Re[z]^2+Im[z]^2; no fancy tricks.
performance-tuning numerics
asked May 10 at 14:17
RomanRoman
8,21811238
$endgroup$
1
$begingroup$
Here's an even slower way: (Re[#]^2 + Im[#]^2) & /@ s. And even slower still: Total[ReIm[#]^2] & /@ s
$endgroup$
– bill s
May 10 at 14:24
add a comment |
$begingroup$
Here I'm not interested in accuracy (see 13614) but rather in raw speed. You'd think that for a complex machine-precision number z, calculating Abs[z]^2 should be faster than calculating Abs[z] because the latter requires a square root whereas the former does not. Yet it's not so:
s = RandomVariate[NormalDistribution[], 10^7, 2].1, I;
Developer`PackedArrayQ[s]
(* True *)
Abs[s]^2; // AbsoluteTiming // First
(* 0.083337 *)
Abs[s]; // AbsoluteTiming // First
(* 0.033179 *)
This indicates that Abs[z]^2 is really calculated by summing the squares of real and imaginary parts, taking a square root (for Abs[z]), and then re-squaring (for Abs[z]^2).
Is there a faster way to compute Abs[z]^2? Is there a hidden equivalent to the GSL's gsl_complex_abs2 function? The source code of this GSL function is simply to return Re[z]^2+Im[z]^2; no fancy tricks.
performance-tuning numerics
asked May 10 at 14:17
RomanRoman
8,21811238
$endgroup$
Here I'm not interested in accuracy (see 13614) but rather in raw speed. You'd think that for a complex machine-precision number z, calculating Abs[z]^2 should be faster than calculating Abs[z] because the latter requires a square root whereas the former does not. Yet it's not so:
s = RandomVariate[NormalDistribution[], 10^7, 2].1, I;
Developer`PackedArrayQ[s]
(* True *)
Abs[s]^2; // AbsoluteTiming // First
(* 0.083337 *)
Abs[s]; // AbsoluteTiming // First
(* 0.033179 *)
This indicates that Abs[z]^2 is really calculated by summing the squares of real and imaginary parts, taking a square root (for Abs[z]), and then re-squaring (for Abs[z]^2).
Is there a faster way to compute Abs[z]^2? Is there a hidden equivalent to the GSL's gsl_complex_abs2 function? The source code of this GSL function is simply to return Re[z]^2+Im[z]^2; no fancy tricks.
performance-tuning numerics
performance-tuning numerics
asked May 10 at 14:17
RomanRoman
8,21811238
asked May 10 at 14:17
RomanRoman
8,21811238
edited May 10 at 14:23
Roman
asked May 10 at 14:17
RomanRoman
8,21811238
asked May 10 at 14:17
RomanRoman
8,21811238
asked May 10 at 14:17
RomanRoman
8,21811238
8,21811238
1
$begingroup$
Here's an even slower way: (Re[#]^2 + Im[#]^2) & /@ s. And even slower still: Total[ReIm[#]^2] & /@ s
$endgroup$
– bill s
May 10 at 14:24
add a comment |
1
$begingroup$
Here's an even slower way: (Re[#]^2 + Im[#]^2) & /@ s. And even slower still: Total[ReIm[#]^2] & /@ s
$endgroup$
– bill s
May 10 at 14:24
1
1
$begingroup$
Here's an even slower way: (Re[#]^2 + Im[#]^2) & /@ s. And even slower still: Total[ReIm[#]^2] & /@ s
$endgroup$
– bill s
May 10 at 14:24
$begingroup$
Here's an even slower way: (Re[#]^2 + Im[#]^2) & /@ s. And even slower still: Total[ReIm[#]^2] & /@ s
$endgroup$
– bill s
May 10 at 14:24
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
There's Internal`AbsSquare:
s = RandomVariate[NormalDistribution[], 10^7, 2].1, I;
foo = Internal`AbsSquare[s]; // AbsoluteTiming // First
murf = Abs[s]^2; // AbsoluteTiming // First
(*
0.022909
0.063441
*)
foo == murf
(* True *)
answered May 10 at 14:24
Michael E2Michael E2
152k12208491
$endgroup$
1
$begingroup$
Ah yes precisely what I was looking for, many thanks Michael! Is there a repository of such tricks?
$endgroup$
– Roman
May 10 at 14:25
1
$begingroup$
@Roman I was just looking. I thought there was a post about usefulInternal`functions, but I couldn't find it just now. The context contains some useful numerical functions likeLog1pandExpm1.Statistics`Library`also contains some nice, well-programmed functions.
$endgroup$
– Michael E2
May 10 at 14:31
2
$begingroup$
@MichaelE2 mathematica.stackexchange.com/questions/805/…
$endgroup$
– Chris K
May 10 at 14:31
1
$begingroup$
@ChrisK That must be it! Thanks.
$endgroup$
– Michael E2
May 10 at 14:32
1
$begingroup$
@CATrevillian I would have thought it was in the MKL (Intel Math Kernel Library), but I didn't find it there. I guess I don't know.
$endgroup$
– Michael E2
May 11 at 3:10
|
show 3 more comments
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
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active
oldest
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active
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votes
$begingroup$
There's Internal`AbsSquare:
s = RandomVariate[NormalDistribution[], 10^7, 2].1, I;
foo = Internal`AbsSquare[s]; // AbsoluteTiming // First
murf = Abs[s]^2; // AbsoluteTiming // First
(*
0.022909
0.063441
*)
foo == murf
(* True *)
answered May 10 at 14:24
Michael E2Michael E2
152k12208491
$endgroup$
1
$begingroup$
Ah yes precisely what I was looking for, many thanks Michael! Is there a repository of such tricks?
$endgroup$
– Roman
May 10 at 14:25
1
$begingroup$
@Roman I was just looking. I thought there was a post about usefulInternal`functions, but I couldn't find it just now. The context contains some useful numerical functions likeLog1pandExpm1.Statistics`Library`also contains some nice, well-programmed functions.
$endgroup$
– Michael E2
May 10 at 14:31
2
$begingroup$
@MichaelE2 mathematica.stackexchange.com/questions/805/…
$endgroup$
– Chris K
May 10 at 14:31
1
$begingroup$
@ChrisK That must be it! Thanks.
$endgroup$
– Michael E2
May 10 at 14:32
1
$begingroup$
@CATrevillian I would have thought it was in the MKL (Intel Math Kernel Library), but I didn't find it there. I guess I don't know.
$endgroup$
– Michael E2
May 11 at 3:10
|
show 3 more comments
$begingroup$
There's Internal`AbsSquare:
s = RandomVariate[NormalDistribution[], 10^7, 2].1, I;
foo = Internal`AbsSquare[s]; // AbsoluteTiming // First
murf = Abs[s]^2; // AbsoluteTiming // First
(*
0.022909
0.063441
*)
foo == murf
(* True *)
answered May 10 at 14:24
Michael E2Michael E2
152k12208491
$endgroup$
1
$begingroup$
Ah yes precisely what I was looking for, many thanks Michael! Is there a repository of such tricks?
$endgroup$
– Roman
May 10 at 14:25
1
$begingroup$
@Roman I was just looking. I thought there was a post about usefulInternal`functions, but I couldn't find it just now. The context contains some useful numerical functions likeLog1pandExpm1.Statistics`Library`also contains some nice, well-programmed functions.
$endgroup$
– Michael E2
May 10 at 14:31
2
$begingroup$
@MichaelE2 mathematica.stackexchange.com/questions/805/…
$endgroup$
– Chris K
May 10 at 14:31
1
$begingroup$
@ChrisK That must be it! Thanks.
$endgroup$
– Michael E2
May 10 at 14:32
1
$begingroup$
@CATrevillian I would have thought it was in the MKL (Intel Math Kernel Library), but I didn't find it there. I guess I don't know.
$endgroup$
– Michael E2
May 11 at 3:10
|
show 3 more comments
$begingroup$
There's Internal`AbsSquare:
s = RandomVariate[NormalDistribution[], 10^7, 2].1, I;
foo = Internal`AbsSquare[s]; // AbsoluteTiming // First
murf = Abs[s]^2; // AbsoluteTiming // First
(*
0.022909
0.063441
*)
foo == murf
(* True *)
answered May 10 at 14:24
Michael E2Michael E2
152k12208491
$endgroup$
There's Internal`AbsSquare:
s = RandomVariate[NormalDistribution[], 10^7, 2].1, I;
foo = Internal`AbsSquare[s]; // AbsoluteTiming // First
murf = Abs[s]^2; // AbsoluteTiming // First
(*
0.022909
0.063441
*)
foo == murf
(* True *)
answered May 10 at 14:24
Michael E2Michael E2
152k12208491
answered May 10 at 14:24
Michael E2Michael E2
152k12208491
answered May 10 at 14:24
Michael E2Michael E2
152k12208491
answered May 10 at 14:24
Michael E2Michael E2
152k12208491
152k12208491
1
$begingroup$
Ah yes precisely what I was looking for, many thanks Michael! Is there a repository of such tricks?
$endgroup$
– Roman
May 10 at 14:25
1
$begingroup$
@Roman I was just looking. I thought there was a post about usefulInternal`functions, but I couldn't find it just now. The context contains some useful numerical functions likeLog1pandExpm1.Statistics`Library`also contains some nice, well-programmed functions.
$endgroup$
– Michael E2
May 10 at 14:31
2
$begingroup$
@MichaelE2 mathematica.stackexchange.com/questions/805/…
$endgroup$
– Chris K
May 10 at 14:31
1
$begingroup$
@ChrisK That must be it! Thanks.
$endgroup$
– Michael E2
May 10 at 14:32
1
$begingroup$
@CATrevillian I would have thought it was in the MKL (Intel Math Kernel Library), but I didn't find it there. I guess I don't know.
$endgroup$
– Michael E2
May 11 at 3:10
|
show 3 more comments
1
$begingroup$
Ah yes precisely what I was looking for, many thanks Michael! Is there a repository of such tricks?
$endgroup$
– Roman
May 10 at 14:25
1
$begingroup$
@Roman I was just looking. I thought there was a post about usefulInternal`functions, but I couldn't find it just now. The context contains some useful numerical functions likeLog1pandExpm1.Statistics`Library`also contains some nice, well-programmed functions.
$endgroup$
– Michael E2
May 10 at 14:31
2
$begingroup$
@MichaelE2 mathematica.stackexchange.com/questions/805/…
$endgroup$
– Chris K
May 10 at 14:31
1
$begingroup$
@ChrisK That must be it! Thanks.
$endgroup$
– Michael E2
May 10 at 14:32
1
$begingroup$
@CATrevillian I would have thought it was in the MKL (Intel Math Kernel Library), but I didn't find it there. I guess I don't know.
$endgroup$
– Michael E2
May 11 at 3:10
1
1
$begingroup$
Ah yes precisely what I was looking for, many thanks Michael! Is there a repository of such tricks?
$endgroup$
– Roman
May 10 at 14:25
$begingroup$
Ah yes precisely what I was looking for, many thanks Michael! Is there a repository of such tricks?
$endgroup$
– Roman
May 10 at 14:25
1
1
$begingroup$
@Roman I was just looking. I thought there was a post about useful
Internal` functions, but I couldn't find it just now. The context contains some useful numerical functions like Log1p and Expm1. Statistics`Library` also contains some nice, well-programmed functions.$endgroup$
– Michael E2
May 10 at 14:31
$begingroup$
@Roman I was just looking. I thought there was a post about useful
Internal` functions, but I couldn't find it just now. The context contains some useful numerical functions like Log1p and Expm1. Statistics`Library` also contains some nice, well-programmed functions.$endgroup$
– Michael E2
May 10 at 14:31
2
2
$begingroup$
@MichaelE2 mathematica.stackexchange.com/questions/805/…
$endgroup$
– Chris K
May 10 at 14:31
$begingroup$
@MichaelE2 mathematica.stackexchange.com/questions/805/…
$endgroup$
– Chris K
May 10 at 14:31
1
1
$begingroup$
@ChrisK That must be it! Thanks.
$endgroup$
– Michael E2
May 10 at 14:32
$begingroup$
@ChrisK That must be it! Thanks.
$endgroup$
– Michael E2
May 10 at 14:32
1
1
$begingroup$
@CATrevillian I would have thought it was in the MKL (Intel Math Kernel Library), but I didn't find it there. I guess I don't know.
$endgroup$
– Michael E2
May 11 at 3:10
$begingroup$
@CATrevillian I would have thought it was in the MKL (Intel Math Kernel Library), but I didn't find it there. I guess I don't know.
$endgroup$
– Michael E2
May 11 at 3:10
|
show 3 more comments
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$begingroup$
Here's an even slower way: (Re[#]^2 + Im[#]^2) & /@ s. And even slower still: Total[ReIm[#]^2] & /@ s
$endgroup$
– bill s
May 10 at 14:24