gonum: gonum.org/v1/gonum/fourier/internal/fftpack

## package fftpack

import "gonum.org/v1/gonum/fourier/internal/fftpack"

Package fftpack implements Discrete Fourier Transform functions ported from the Fortran implementation of FFTPACK.

### func Cfftb¶Uses

func Cfftb(n int, r, work []float64, ifac []int)

Cfftb computes the backward complex Discrete Fourier Transform (the Fourier synthesis). Equivalently, Cfftf computes the computes a complex periodic sequence from its Fourier coefficients. The transform is defined below at output parameter c.

Input parameters:

n      The length of the array c to be transformed. The method
is most efficient when n is a product of small primes.
n may change so long as different work arrays are provided.

c      A complex array of length n which contains the sequence
to be transformed.

work   A real work array which must be dimensioned at least 4*n.
in the program that calls Cfftb. The work array must be
initialized by calling subroutine Cffti(n,work,ifac) and a
different work array must be used for each different
value of n. This initialization does not have to be
repeated so long as n remains unchanged thus subsequent
transforms can be obtained faster than the first.
The same work array can be used by Cfftf and Cfftb.

ifac   A work array containing the factors of n. ifac must have
length of at least 15.

Output parameters:

c      for j=0, ..., n-1
c[j]=the sum from k=0, ..., n-1 of
c[k]*exp(i*j*k*2*pi/n)

where i=sqrt(-1)

This transform is unnormalized since a call of Cfftf
followed by a call of Cfftb will multiply the input
sequence by n.

The n elements of c are represented in n pairs of real
values in r where c[j] = r[j*2]+r[j*2+1]i.

work   Contains results which must not be destroyed between
calls of Cfftf or Cfftb.
ifac   Contains results which must not be destroyed between
calls of Cfftf or Cfftb.

### func Cfftf¶Uses

func Cfftf(n int, r, work []float64, ifac []int)

Cfftf computes the forward complex Discrete Fourier transform (the Fourier analysis). Equivalently, Cfftf computes the Fourier coefficients of a complex periodic sequence. The transform is defined below at output parameter c.

Input parameters:

n      The length of the array c to be transformed. The method
is most efficient when n is a product of small primes.
n may change so long as different work arrays are provided.

c      A complex array of length n which contains the sequence
to be transformed.

work   A real work array which must be dimensioned at least 4*n.
in the program that calls Cfftf. The work array must be
initialized by calling subroutine Cffti(n,work,ifac) and a
different work array must be used for each different
value of n. This initialization does not have to be
repeated so long as n remains unchanged thus subsequent
transforms can be obtained faster than the first.
the same work array can be used by Cfftf and Cfftb.

ifac   A work array containing the factors of n. ifac must have
length of at least 15.

Output parameters:

c     for j=0, ..., n-1
c[j]=the sum from k=0, ..., n-1 of
c[k]*exp(-i*j*k*2*pi/n)

where i=sqrt(-1)

This transform is unnormalized since a call of Cfftf
followed by a call of Cfftb will multiply the input
sequence by n.

The n elements of c are represented in n pairs of real
values in r where c[j] = r[j*2]+r[j*2+1]i.

work   Contains results which must not be destroyed between
calls of Cfftf or Cfftb.
ifac   Contains results which must not be destroyed between
calls of Cfftf or Cfftb.

### func Cffti¶Uses

func Cffti(n int, work []float64, ifac []int)

Cffti initializes the array work which is used in both Cfftf and Cfftb. the prime factorization of n together with a tabulation of the trigonometric functions are computed and stored in work.

input parameter

n      The length of the sequence to be transformed.

Output parameters:

work   A work array which must be dimensioned at least 4*n.
the same work array can be used for both Cfftf and Cfftb
as long as n remains unchanged. Different work arrays
are required for different values of n. The contents of
work must not be changed between calls of Cfftf or Cfftb.

ifac   A work array containing the factors of n. ifac must have
length 15.

### func Cosqb¶Uses

func Cosqb(n int, x, work []float64, ifac []int)

Cosqb computes the Fast Fourier Transform of quarter wave data. That is, Cosqb computes a sequence from its representation in terms of a cosine series with odd wave numbers. The transform is defined below at output parameter x.

Cosqf is the unnormalized inverse of Cosqb since a call of Cosqb followed by a call of Cosqf will multiply the input sequence x by 4*n.

The array work which is used by subroutine Cosqb must be initialized by calling subroutine Cosqi(n,work).

#### Input parameters ¶

n The length of the array x to be transformed. The method

is most efficient when n is a product of small primes.

x An array which contains the sequence to be transformed.

work A work array which must be dimensioned at least 3*n

in the program that calls Cosqb. The work array must be
initialized by calling subroutine Cosqi(n,work) and a
different work array must be used for each different
value of n. This initialization does not have to be
repeated so long as n remains unchanged thus subsequent
transforms can be obtained faster than the first.

ifac An integer work array of length at least 15.

#### Output parameters ¶

x for i=0, ..., n-1

x[i]= the sum from k=0 to k=n-1 of
4*x[k]*cos((2*k+1)*i*pi/(2*n))

A call of Cosqb followed by a call of
Cosqf will multiply the sequence x by 4*n.
Therefore Cosqf is the unnormalized inverse
of Cosqb.

work Contains initialization calculations which must not

be destroyed between calls of Cosqb or Cosqf.

### func Cosqf¶Uses

func Cosqf(n int, x, work []float64, ifac []int)

Cosqf computes the Fast Fourier Transform of quarter wave data. That is, Cosqf computes the coefficients in a cosine series representation with only odd wave numbers. The transform is defined below at output parameter x.

Cosqb is the unnormalized inverse of Cosqf since a call of Cosqf followed by a call of Cosqb will multiply the input sequence x by 4*n.

The array work which is used by subroutine Cosqf must be initialized by calling subroutine Cosqi(n,work).

#### Input parameters ¶

n The length of the array x to be transformed. The method

is most efficient when n is a product of small primes.

x An array which contains the sequence to be transformed.

work A work array which must be dimensioned at least 3*n

in the program that calls Cosqf. The work array must be
initialized by calling subroutine Cosqi(n,work) and a
different work array must be used for each different
value of n. This initialization does not have to be
repeated so long as n remains unchanged thus subsequent
transforms can be obtained faster than the first.

ifac An integer work array of length at least 15.

#### Output parameters ¶

x for i=0, ..., n-1

x[i] = x[i] + the sum from k=0 to k=n-2 of
2*x[k]*cos((2*i+1)*k*pi/(2*n))

A call of Cosqf followed by a call of
Cosqb will multiply the sequence x by 4*n.
Therefore Cosqb is the unnormalized inverse
of Cosqf.

work Contains initialization calculations which must not

be destroyed between calls of Cosqf or Cosqb.

### func Cosqi¶Uses

func Cosqi(n int, work []float64, ifac []int)

Cosqi initializes the array work which is used in both Cosqf and Cosqb. The prime factorization of n together with a tabulation of the trigonometric functions are computed and stored in work.

#### Input parameter ¶

n The length of the sequence to be transformed. the method

is most efficient when n+1 is a product of small primes.

#### Output parameters ¶

work A work array which must be dimensioned at least 3*n.

The same work array can be used for both Cosqf and Cosqb
as long as n remains unchanged. Different work arrays
are required for different values of n. The contents of
work must not be changed between calls of Cosqf or Cosqb.

ifac An integer work array of length at least 15.

### func Cost¶Uses

func Cost(n int, x, work []float64, ifac []int)

Cost computes the Discrete Fourier Cosine Transform of an even sequence x(i). The transform is defined below at output parameter x.

Cost is the unnormalized inverse of itself since a call of Cost followed by another call of Cost will multiply the input sequence x by 2*(n-1). The transform is defined below at output parameter x

The array work which is used by subroutine Cost must be initialized by calling subroutine Costi(n,work).

#### Input parameters ¶

n The length of the sequence x. n must be greater than 1.

The method is most efficient when n-1 is a product of
small primes.

x An array which contains the sequence to be transformed.

work A work array which must be dimensioned at least 3*n

in the program that calls Cost. The work array must be
initialized by calling subroutine Costi(n,work) and a
different work array must be used for each different
value of n. This initialization does not have to be
repeated so long as n remains unchanged thus subsequent
transforms can be obtained faster than the first.

ifac An integer work array of length at least 15.

#### Output parameters ¶

x for i=1,...,n

x(i) = x(1)+(-1)**(i-1)*x(n)
+ the sum from k=2 to k=n-1
2*x(k)*cos((k-1)*(i-1)*pi/(n-1))

A call of Cost followed by another call of
Cost will multiply the sequence x by 2*(n-1).
Hence Cost is the unnormalized inverse
of itself.

work Contains initialization calculations which must not be

destroyed between calls of Cost.

ifac An integer work array of length at least 15.

### func Costi¶Uses

func Costi(n int, work []float64, ifac []int)

Costi initializes the array work which is used in subroutine Cost. The prime factorization of n together with a tabulation of the trigonometric functions are computed and stored in work.

#### Input parameter ¶

n The length of the sequence to be transformed. The method

is most efficient when n-1 is a product of small primes.

#### Output parameters ¶

work A work array which must be dimensioned at least 3*n.

Different work arrays are required for different values
of n. The contents of work must not be changed between
calls of Cost.

ifac An integer work array of length at least 15.

### func Rfftb¶Uses

func Rfftb(n int, r, work []float64, ifac []int)

Rfftb computes the real perodic sequence from its Fourier coefficients (Fourier synthesis). The transform is defined below at output parameter r.

Input parameters

n      The length of the array r to be transformed. The method
is most efficient when n is a product of small primes.
n may change so long as different work arrays are provided.

r      A real array of length n which contains the sequence
to be transformed.

work   A work array which must be dimensioned at least 2*n.
in the program that calls Rfftb. The work array must be
initialized by calling subroutine rffti(n,work,ifac) and a
different work array must be used for each different
value of n. This initialization does not have to be
repeated so long as n remains unchanged thus subsequent
transforms can be obtained faster than the first.
The same work array can be used by Rfftf and Rfftb.

ifac   A work array containing the factors of n. ifac must have
length of at least 15.

output parameters

r      for n even and for i = 0, ..., n
r[i] = r[0]+(-1)^i*r[n-1]
plus the sum from k=1 to k=n/2-1 of
2*r(2*k-1)*cos(k*i*2*pi/n)
-2*r(2*k)*sin(k*i*2*pi/n)

for n odd and for i = 0, ..., n-1
r[i] = r[0] plus the sum from k=1 to k=(n-1)/2 of
2*r(2*k-1)*cos(k*i*2*pi/n)
-2*r(2*k)*sin(k*i*2*pi/n)

This transform is unnormalized since a call of Rfftf
followed by a call of Rfftb will multiply the input
sequence by n.

work   Contains results which must not be destroyed between
calls of Rfftf or Rfftb.
ifac   Contains results which must not be destroyed between
calls of Rfftf or Rfftb.

### func Rfftf¶Uses

func Rfftf(n int, r, work []float64, ifac []int)

Rfftf computes the Fourier coefficients of a real perodic sequence (Fourier analysis). The transform is defined below at output parameter r.

Input parameters:

n      The length of the array r to be transformed. The method
is most efficient when n is a product of small primes.
n may change so long as different work arrays are provided.

r      A real array of length n which contains the sequence
to be transformed.

work   a work array which must be dimensioned at least 2*n.
in the program that calls Rfftf. the work array must be
initialized by calling subroutine rffti(n,work,ifac) and a
different work array must be used for each different
value of n. This initialization does not have to be
repeated so long as n remains unchanged. Thus subsequent
transforms can be obtained faster than the first.
The same work array can be used by Rfftf and Rfftb.

ifac   A work array containing the factors of n. ifac must have
length of at least 15.

Output parameters:

r      r[0] = the sum from i=0 to i=n-1 of r[i]

if n is even set l=n/2, if n is odd set l = (n+1)/2
then for k = 1, ..., l-1
r[2*k-1] = the sum from i = 0 to i = n-1 of
r[i]*cos(k*i*2*pi/n)
r[2*k] = the sum from i = 0 to i = n-1 of
-r[i]*sin(k*i*2*pi/n)

if n is even
r[n-1] = the sum from i = 0 to i = n-1 of
(-1)^i*r[i]

This transform is unnormalized since a call of Rfftf
followed by a call of Rfftb will multiply the input
sequence by n.

work   contains results which must not be destroyed between
calls of Rfftf or Rfftb.
ifac   contains results which must not be destroyed between
calls of Rfftf or Rfftb.

### func Rffti¶Uses

func Rffti(n int, work []float64, ifac []int)

Rffti initializes the array work which is used in both Rfftf and Rfftb. The prime factorization of n together with a tabulation of the trigonometric functions are computed and stored in work.

Input parameter:

n      The length of the sequence to be transformed.

Output parameters:

work   A work array which must be dimensioned at least 2*n.
The same work array can be used for both Rfftf and Rfftb
as long as n remains unchanged. different work arrays
are required for different values of n. The contents of
work must not be changed between calls of Rfftf or Rfftb.

ifac   A work array containing the factors of n. ifac must have
length of at least 15.

### func Sinqb¶Uses

func Sinqb(n int, x, work []float64, ifac []int)

Sinqb computes the Fast Fourier Transform of quarter wave data. That is, Sinqb computes a sequence from its representation in terms of a sine series with odd wave numbers. The transform is defined below at output parameter x.

Sinqf is the unnormalized inverse of Sinqb since a call of Sinqb followed by a call of Sinqf will multiply the input sequence x by 4*n.

The array work which is used by subroutine Sinqb must be initialized by calling subroutine Sinqi(n,work).

#### Input parameters ¶

n The length of the array x to be transformed. The method

is most efficient when n is a product of small primes.

x An array which contains the sequence to be transformed.

work A work array which must be dimensioned at least 3*n.

in the program that calls Sinqb. The work array must be
initialized by calling subroutine Sinqi(n,work) and a
different work array must be used for each different
value of n. This initialization does not have to be
repeated so long as n remains unchanged thus subsequent
transforms can be obtained faster than the first.

ifac An integer work array of length at least 15.

#### Output parameters ¶

x for i=0, ..., n-1

x[i]= the sum from k=0 to k=n-1 of
4*x[k]*sin((2*k+1)*i*pi/(2*n))

A call of Sinqb followed by a call of
Sinqf will multiply the sequence x by 4*n.
Therefore Sinqf is the unnormalized inverse
of Sinqb.

work Contains initialization calculations which must not

be destroyed between calls of Sinqb or Sinqf.

### func Sinqf¶Uses

func Sinqf(n int, x, work []float64, ifac []int)

Sinqf computes the Fast Fourier Transform of quarter wave data. That is, Sinqf computes the coefficients in a sine series representation with only odd wave numbers. The transform is defined below at output parameter x.

Sinqb is the unnormalized inverse of Sinqf since a call of Sinqf followed by a call of Sinqb will multiply the input sequence x by 4*n.

The array work which is used by subroutine Sinqf must be initialized by calling subroutine Sinqi(n,work).

#### Input parameters ¶

n The length of the array x to be transformed. The method

is most efficient when n is a product of small primes.

x An array which contains the sequence to be transformed.

work A work array which must be dimensioned at least 3*n.

in the program that calls Sinqf. The work array must be
initialized by calling subroutine Sinqi(n,work) and a
different work array must be used for each different
value of n. This initialization does not have to be
repeated so long as n remains unchanged thus subsequent
transforms can be obtained faster than the first.

ifac An integer work array of length at least 15.

#### Output parameters ¶

x for i=0, ..., n-1

x[i] = (-1)^(i)*x[n-1]
+ the sum from k=0 to k=n-2 of
2*x[k]*sin((2*i+1)*k*pi/(2*n))

A call of Sinqf followed by a call of
Sinqb will multiply the sequence x by 4*n.
Therefore Sinqb is the unnormalized inverse
of Sinqf.

work Contains initialization calculations which must not

be destroyed between calls of Sinqf or Sinqb.

### func Sinqi¶Uses

func Sinqi(n int, work []float64, ifac []int)

Sinqi initializes the array work which is used in both Sinqf and Sinqb. The prime factorization of n together with a tabulation of the trigonometric functions are computed and stored in work.

#### Input parameter ¶

n The length of the sequence to be transformed. The method

is most efficient when n+1 is a product of small primes.

#### Output parameter ¶

work A work array which must be dimensioned at least 3*n.

The same work array can be used for both Sinqf and Sinqb
as long as n remains unchanged. Different work arrays
are required for different values of n. The contents of
work must not be changed between calls of Sinqf or Sinqb.

ifac An integer work array of length at least 15.

### func Sint¶Uses

func Sint(n int, x, work []float64, ifac []int)

Sint computes the Discrete Fourier Sine Transform of an odd sequence x(i). The transform is defined below at output parameter x.

Sint is the unnormalized inverse of itself since a call of Sint followed by another call of Sint will multiply the input sequence x by 2*(n+1).

The array work which is used by subroutine Sint must be initialized by calling subroutine Sinti(n,work).

#### Input parameters ¶

n The length of the sequence to be transformed. The method

is most efficient when n+1 is the product of small primes.

x An array which contains the sequence to be transformed.

work A work array with dimension at least ceil(2.5*n)

in the program that calls Sint. The work array must be
initialized by calling subroutine Sinti(n,work) and a
different work array must be used for each different
value of n. This initialization does not have to be
repeated so long as n remains unchanged thus subsequent
transforms can be obtained faster than the first.

ifac An integer work array of length at least 15.

#### Output parameters ¶

x for i=1,...,n

x(i)= the sum from k=1 to k=n
2*x(k)*sin(k*i*pi/(n+1))

A call of Sint followed by another call of
Sint will multiply the sequence x by 2*(n+1).
Hence Sint is the unnormalized inverse
of itself.

work Contains initialization calculations which must not be

destroyed between calls of Sint.

ifac Contains initialization calculations which must not be

destroyed between calls of Sint.

### func Sinti¶Uses

func Sinti(n int, work []float64, ifac []int)

Sinti initializes the array work which is used in subroutine Sint. The prime factorization of n together with a tabulation of the trigonometric functions are computed and stored in work.

#### Input parameter ¶

n The length of the sequence to be transformed. The method

is most efficient when n+1 is a product of small primes.

#### Output parameter ¶

work A work array with at least ceil(2.5*n) locations.

Different work arrays are required for different values
of n. The contents of work must not be changed between
calls of Sint.

ifac An integer work array of length at least 15.

Package fftpack imports 1 packages (graph) and is imported by 1 packages. Updated 2019-04-05. Refresh now. Tools for package owners.