我遇到了计算Fortran中某些数据的fft的问题。我不知道算法是否有问题,舍入,缺乏精确度或者是什么。
这是代码
SEXP RODBCcheckchannel(SEXP chan, SEXP id)
{
SEXP ptr = getAttrib(chan, install("handle_ptr"));
pRODBCHandle thisHandle = R_ExternalPtrAddr(ptr);
return ScalarLogical(thisHandle && TYPEOF(ptr) == EXTPTRSXP &&
thisHandle->channel == asInteger(chan) &&thisHandle->id == asInteger(id));
}
在fortran中运行
module fft_mod
public :: fft1D
integer,parameter :: cp = selected_real_kind(14)
real(cp),parameter :: PI = real(3.14159265358979,cp)
contains
subroutine fft1D(x)
complex(cp), dimension(:), intent(inout) :: x
complex(cp), dimension(:), allocatable :: temp
complex(cp) :: S
integer :: N
complex(cp) :: j ! sqrt(-1)
integer :: i,k
N=size(x)
allocate(temp(N))
j = cmplx(0.0_cp,1.0_cp,cp)
S = cmplx(0.0_cp,0.0_cp,cp)
temp = cmplx(0.0_cp,0.0_cp,cp)
do i = 1,N
do k = 1,N
S = S + x(k)*exp(real(-2.0,cp)*PI*j*real(k-1,cp)*real(i-1,cp)/real(N,cp))
enddo
temp(i) = S
S = cmplx(0.0_cp,0.0_cp,cp)
enddo
x = temp
deallocate(temp)
end subroutine
end module
program test
use fft_mod
implicit none
complex(cp), dimension(10) :: data = (/1.0, 2.0, 3.0, 4.0, 5.0, 5.0, 4.0, 3.0, 2.0, 1.0/)
integer :: i
call fft1D(data)
do i=1,10
write(*,*) data(i)
end do
end program test
但是在matlab中运行相同的数据集会给出
C:\Users\Charlie\Desktop\fft>gmake
gfortran -J".\\mod" -fimplicit-none -Wuninitialized -g -Wall -Wextra -fbacktrace
-fcheck=all -O0 -fopenmp -D_QUAD_PRECISION_ -cpp -c -o .\\obj\testFFT.o testFFT
.f90
gfortran -o .\\test -J".\\mod" -fimplicit-none -Wuninitialized -g -Wall -Wextra
-fbacktrace -fcheck=all -O0 -fopenmp -D_QUAD_PRECISION_ -cpp .\\obj\testFFT.o
C:\Users\Charlie\Desktop\fft>test.exe
( 30.000000000000000 , 0.0000000000000000 )
( -9.4721355260035178 , -3.0776825738331275 )
( 1.2032715918097736E-007, 8.7422769579070803E-008)
(-0.52786408204828272 ,-0.72654221835813126 )
( 5.6810824045072650E-008, 1.7484555003832725E-007)
( 1.0325074129013956E-013, 2.6226834001115759E-007)
( -8.5216018574918451E-008, 2.6226836247200680E-007)
(-0.52786395855490920 , 0.72654325051559143 )
( -4.8130813040669906E-007, 3.4969128892559098E-007)
( -9.4721398159606647 , 3.0776922072585111 )
我相信我使用selected_real_kind(14)使用双精度,但看起来结果只是单精度。我确信一些散布的真实(,cp)不是必需的,但我不知道在matlab中,结果是单精度的地方,方式或原因。
非常感谢任何帮助!
更新:
使用已接受答案的建议,唯一需要改变的是:
format long ; g = [1:5 5:-1:1]; fft(g)'
ans =
30.000000000000000
-9.472135954999580 + 3.077683537175253i
0
-0.527864045000420 + 0.726542528005361i
0
0
0
-0.527864045000420 - 0.726542528005361i
0
-9.472135954999580 - 3.077683537175253i
到
real(cp),parameter :: PI = real(3.14159265358979,cp)
答案 0 :(得分:2)
问题在于如何定义实数,特别是pi
。定义时
real(cp),parameter :: PI = real(3.14159265358979,cp)
您将参数3.14159265358979传递给函数real
。但实数具有默认的单精度,因此您的实数在进入函数时会被转换为单精度。请考虑以下示例:
program main
integer,parameter :: cp = selected_real_kind(14)
real(cp),parameter :: pi = real(3.14159265358979,cp)
real(cp),parameter :: pj = 3.14159265358979_cp
write(*,*) pi
write(*,*) pj
end program main
使用pgfortran
编译而没有选项,这给了我:
3.141592741012573
3.141592653589790
在定义任何实数时,您应使用[]_cp
来指定种类,而不是real([],cp)
。
修改:此问题也会影响您定义0.0
,1.0
和2.0
的方式,但这些数字可能会完全转换为二进制,并且不会出现相同的舍入错误。< / p>
答案 1 :(得分:0)
如果你只需要“接近”双精度(Real(8)),那么上面接受的答案是一个合理的解决方案,因为你已经明确地将Pi定义为接近Real(8)的数字。 Pi到完全Real(8)的实际值是
3.1415926535897931
c.f。你的参数
3.14159265358979
如果您希望获得更一般的“与_cp一致的完全精度”结果,您可能希望使用类似
的内容Real(cP), Parameter :: Pi = 4_cp*ATan(1_cp)
***编辑:感谢francescalus发现拼写错误,这应该是ATan(1._cp),尽管如此,如下所述,应该使用“参数化”数字并避免“_cp”一起使用args等。
现在,如果_cp是16,你的Pi将会自动:
3.14159265358979323846264338327950280
现在,每当您更改_cp时,您的代码将自动具有“与_cp一致的完全精度”。
顺便说一句,您也可以希望“参数化”某些基本数字,例如:Real(cp), Parameter :: Zero = 0_cp
Real(cp), Parameter :: One = 1_cp
:
...等
***编辑:感谢francescalus发现拼写错误,但在这些特殊的表达中,虽然0.0_cp和1.0_cp可能会更好,但是没有关系,因为他们Params声明处理了&gt;
然后,在您的代码中,您可以编写,例如:
...
Real(cp), Parameter :: Pi = Four*ATan(One)
...
S = Cmplx(Zero, Zero)*Exp(-Two) ! etc etc
并且不必担心在整个地方添加_cp等,并且更容易阅读/调试等。
此策略具有某些其他优势,特别是在遗留代码中,例如
If( a == 1.0 ) Then
VS
If( a == One ) Then
......但这是另一个故事。
顺便提一下,在某种程度上,“风格”的问题,在Fortran中,算术默认为表达式中的“最大”精度/类型,所以
S = S + x(k)*exp(real(-2.0,cp)*PI*j*real(k-1,cp)*real(i-1,cp)/real(N,cp))
应该相当于
S = S + x(k)*exp(-2*PI*j*(k-1)*(i-1)/N)
或者,甚至更好
S = S + x(k)*exp(-Two*PI*j*(k-1)*(i-1)/N)
就个人而言,我发现阅读越容易,就越容易做到。
...尽管如上所述,如果您的算术仅包含整数,则可能需要考虑其他因素。但是,如果你“参数化”只有Real,那么你应该永远不会有使用“2”的风险(例如div / 0),如果你使用“2”则需要真正的2.0。
最后,因为你有“参数化”的Pi,为什么不用2 * Pi,例如:
...
Real(cp), Parameter :: TwoPi = Two*Pi
现在,表达式
S = S + x(k)*exp(-Two*PI*j*(k-1)*(i-1)/N)
可以
S = S + x(k)*exp(-TwoPI*j*(k-1)*(i-1)/N)
节省了一点CPU。实际上,扩展此逻辑可以进一步提高代码的性能。
...在一个完全独立的点上,你知道你的var Temp()可以是“自动”,比如说,
complex(cp) :: temp(Size(x, Dim = 1))
然后不需要Allocatable等。虽然从我坐的地方看起来可能完全没有“S”和“Temp(:)”编写你的代码,提供了更简单和更快的代码。
*****附录:根据我的回答中的一些评论,我认为可能有助于显示OP代码的可能更改以提高性能,并在某种程度上进一步提高精度。
然而,在此之前,OP没有说明为什么需要特定的精确度。在许多工业/现实世界环境中,这可能是一个重要问题。例如,在某些情况下,与精度相比,速度更为重要,只要精度足以提供特定于上下文的可靠性/决策制定。例如,可以想象在某些情况下,计算应该全部为Real(4)。这个问题需要一个单独的,非平凡的讨论。
尽管如此,罗斯的答案在“记忆表现”方面纠正了精确度问题,而我的原始答案在“你实际上需要提供足够的sig figs开头”方面提高了精确度(即使声明是正确的),减少FLOPS的数量不仅可以提高速度,还可以提高精度,因为每个FLOP都会引入截断/舍入的可能性,特别是对于单个表达式中的长序列FLOP。此外,这个OP问题可以通过“身份”公式很好地解决(4 * ATan(1))。在许多情况下,这是不可能或不实用的,然后需要一些不太优雅的方法,但我将这些方法留了一天。
以下只是一两个可能的替代方案。
因此可能需要调整;请随时提出改进建议。
运气好的话,它可能会使FLOP减少约80%左右。
...如果方便的话,我会感谢OP对不同的实现进行基准测试并报告结果。您可能还希望对_cp = 4,8,16进行基准测试,以证明速度和精度之间的权衡。
这个替代版本需要明显更新呼叫s / r。
module fft_mod
!
public :: fft1D
!
Integer,parameter :: cp = selected_real_kind(14) ! just out of curiosity, why "14", see my comments on "reality vs. precision"
!>>real(cp),parameter :: PI = real(3.14159265358979,cp)
!
Real(cp), Parameter :: Zero = 0_cp ! these types of declarations can be created in a separate Module,
! say "MyDeclarationModule", and then just access with "Use"
!
!XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
! NOTE: With respect to francescalus comments on issues with "1_cp" syntax, the usage here works fine as shown in the previous result,
! though francescalus' comments are fair in suggesting that other approaches make for better coding.
! However, and to be clear, I don't actually use "_xx" style myslef. I have used it here ONLY to be consistent with the
! the OP's and earlier answers. The usage of it here is NOT a recommendation. A discussion of the alternatives
! is much too long and strays from the immediate issues too far.
! ... Perhaps francescalus will take up the mantle and re-write the OP's code for alternatives
!XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
!
Real(cp), Parameter :: One = 1_cp
Real(cp), Parameter :: Two = 2_cp
Real(cp), Parameter :: Four = 4_cp
!
! ... after this point, there is no need to see "_cp" again, in this example
!
Real(cp), Parameter :: Pi = Four*ATan(One) ! this guarrantees maximal precision for Pi, up to "_cp"
!
Real(cp), Parameter :: TwoPi = Two*Pi ! Vladimir: now, and only now (that I've talem most everything out of the loop),
! this declaration has less value ... unlike previously, when it was
! still in the inner NxN, and when it saved approx 15 - 20% of FLOPs
! Crucially, if you do a lot of computational mathematics, TwoPi, PiBy2, Root2Pi, etc etc
! arise with considerable frequency, and creating these as Parameters makes for much
! improvement, and I would think it a practice to be encouraged.
!
Complex(cp), Parameter :: SqrtM1 = Cmplx(Zero, One) ! sqrt(-1), this is your "j",
! sorry but "j" sounds just too much like an "integer" to me
! (for us "old timers"), so I changed the name to something more meaningful to me
!
!
Contains
!
Pure Subroutine fft1D(x, n, Temp) ! OPTIONAL, return the results explicitly to save vector assignment, and preserve original data
! OPTIONAL, send the size of x() explicitly as n, much cleaner
! you could leave it as s/r fft1D(x), and most of the savings below would still apply
! ... also, my practice is to make everything Pure/Elemental where reasonable
!
Integer, Intent(In) :: n ! Optional, but cleaner
! complex(cp), Intent(InOut) :: x(n) ! consider as just Intent(In), and return Temp() as Intent(Out)
complex(cp), Intent(In) :: x(n) ! consider as just Intent(In), and return Temp() as Intent(Out)
!
! Locals
! ------
!
! Complex(cp), Allocatable :: temp(:) ! no need for Allocatable
! Complex(cp) :: temp(Size(x), Dim = 1) ! I prefer to pass n explicitly
Complex(cp), Intent(Out) :: temp(n) ! make Intent(Out) to preserve data and save a vector assignment
!
! complex(cp) :: S
! integer :: N
! complex(cp) :: j ! sqrt(-1)
!
Integer :: i, k
!
Complex(cp) :: TPSdivN ! new reduce 4 nxn mults to 1
!
Complex(cp) :: TPSdivNiM1 ! new, to reduce 5 nxn mults to 1 nx1
!
Real(cp) :: rKM1(n) ! new, to obviate nxn k-1's in inner loop
!
! N=size(x) ! my preference is to pass N explicitly,
! but can be done this way if prefered (need to make appropriate changes above)
!
! allocate(temp(N)) ! Temp() can be either automatic or an Arg, no need for Allocate
! j = cmplx(0.0_cp,1.0_cp,cp) ! make Parameter, and rename to SqrtM1 (personal pref)
! S = cmplx(0.0_cp,0.0_cp,cp) ! Not required with "improved" inner loop
!
!
temp(1:n) = Cmplx(Zero, Zero) ! S is not needed, just start with Temp() directly
!
TPSdivN = -TwoPi*SqrtM1/n ! new, move everything out all loops that can be
!
ForAll( k=1:n) rKM1(k) = k - 1 ! new, this allows the elimination of NxN "k-1's" in the inner
! my preference is for ForAll's, but you could use Do,
!
do i = 1,N
!
TPSdivNiM1 = TPSdivN*(i-1) ! new. everything out of inner loop that can be
!
! improved, inner do, but could use "no-Do" alternative as illustrated below
!
! do k = 1,N
!>> S = S + x(k)*exp(real(-2.0,cp)*PI*j*real(k-1,cp)*real(i-1,cp)/real(N,cp)) ! original, many unneccessary nxn FLOPs, type conversions, etc
! temp(i) = temp(i) + x(k)*Exp(TPSdivNiM1*rKM1(k))) ! use array of k-1's, then the inner k-loop is no longer required,
! and can use Product()/Sum() or Dot_Product() intrinsics, see example below
! End Do
!
! alternative "direct array" approach to inner loop, or "no-DO" version ... there are other possibilities.
!
Temp(i) = Dot_Product( x(1:n), Exp(TPSdivNiM1*rKM1(1:n)) ) ! this approach can have certain advantages depending on circumstances
!
! temp(i) = S ! not needed
! S = cmplx(0.0_cp,0.0_cp,cp) ! not needed
!
End Do
!
! x(1:n) = temp(1:n) ! not need if Temp() is Intent(Out) that save this vector assignment, and the original data
! deallocate(temp) ! not needed
!
End Subroutine fft1D
!
! end module
End Module fft_mod