◆ Sinqmf()

One-dimensional quarter sine transform (multiple sequences)

Purpose: This routine computes the one-dimensional Fourier transform of multiple sequences within a real array, where each of a sequence is a sine series with odd wave numbers. This is referred to as the forward transform or Fourier analysis, transforming the sequence from physical to spectral space.
R(l*jump+k) = (2/N)ΣR(l*jump+j)sin(π(j+1)(2k+1)/(2n)) + (-1)^(k+2)R(l*jump+N-1)/N (Σ for j = 0 to N-2) (l = 0 to lot-1, k = 0 to N-1)

This transform is normalized since a call to Sinqmf followed by a call to Sinqmb (or vice-versa) reproduces the original array subject to algorithmic constraints, roundoff error, etc.

Parameters

[in]	Lot	Number of sequences to be transformed. (Lot >= 1)
[in]	Jump	Increment between the locations, in array R(), of the first elements of two consecutive sequences to be transformed. (Jump >= 1)
[in]	N	Length of the sequence to be transformed. (N >= 1) (The transform is most efficient when N is a product of small primes)
[in,out]	R()	Array R(LR - 1) (LR >= (Lot - 1)Jump + Inc(N - 1) + 1) [in] The sequences to be transformed. [out] The Fourier forward transformed sequences of data.
[in]	Wsave()	Array Wsave(LWsave - 1) (LWsave >= 2*N + ln(N)/ln(2) + 4) Work data. Its contents must be initialized with a call to Sinqmi before the first call to Sinqmf or Sinqmb for a given transform length N.
[out]	Info	= 0: Successful exit. = -1: The argument Lot had an illegal value. (Lot < 1, or, Lot, Jump, N and Inc are inconsistent) = -2: The argument Jump had an illegal value. (Jump < 1) = -3: The argument N had an illegal value. (N < 1) = -4: The argument R() had an illegal value. (Array R() is not big enough) = -5: The argument Wsave() had an illegal value. (Array Wsave() is not big enough) = -7: The argument Inc had an illegal value. (Inc < 1)
[in]	Inc	(Optional) Integer increment between the locations, in array R(), of two consecutive elements within the sequence. (Inc >= 1) (default = 1)

Example Program: Compute the sine transform and backward transform of 2 sequences of 5 random data successively, and compare with the original data sequences.
Sub Ex_Sinqm()

Const N = 5, Lot = 2, Jump = N

Dim Wsave() As Double, R(Lot * N - 1) As Double, R0(Lot * N - 1) As Double

Dim LWsave As Long, Info As Long, I As Long, J As Long, K As Long

'-- Initialization

LWsave = 2 * N + Log(N) / Log(2) + 4

ReDim Wsave(LWsave - 1)

Call Sinqmi(N, Wsave, Info)

If Info <> 0 Then GoTo Err

'-- Generate test data

For I = 0 To Lot * N - 1

R(I) = Rnd()

R0(I) = R(I)

Next

'-- Forward transform

Call Sinqmf(Lot, Jump, N, R(), Wsave(), Info)

If Info <> 0 Then GoTo Err

'-- Backward transform

Call Sinqmb(Lot, Jump, N, R(), Wsave(), Info)

If Info <> 0 Then GoTo Err

'-- Print result

For J = 0 To Lot - 1

For I = 0 To N - 1

K = J * Jump + I

Debug.Print R0(K), R(K), R(K) - R0(K)

Next

Debug.Print

Next

Exit Sub

Err:

Debug.Print "Error in Sinqmi/Sinqmf/Sinqmb: Info =", Info

End Sub

Example Results: 0.803530812263489 0.803530812263489 0

0.63199120759964 0.63199120759964 0

4.37657833099365E-02 4.37657833099366E-02 5.55111512312578E-17

0.804474771022797 0.804474771022797 0

0.471749663352966 0.471749663352966 1.11022302462516E-16

0.809219419956207 0.809219419956207 2.22044604925031E-16

0.216426849365234 0.216426849365234 1.11022302462516E-16

0.423454463481903 0.423454463481903 1.11022302462516E-16

0.453921914100647 0.453921914100647 1.66533453693773E-16

0.595368683338165 0.595368683338165 1.11022302462516E-16