C06 Chapter Introduction : NAG Library, Mark 27

This chapter is concerned with the following tasks.

(a)Calculating the discrete Fourier transform of a sequence of real or complex data values.
(b)Calculating the discrete convolution or the discrete correlation of two sequences of real or complex data values using discrete Fourier transforms.
(c)Calculating the inverse Laplace transform of a user-supplied subroutine.
(d)Calculating the fast Gauss transform approximation to the discrete Gauss transform.
(e)Direct summation of orthogonal series.
(f)Acceleration of convergence of a sequence of real values.

Most of the routines in this chapter calculate the finite discrete Fourier transform (DFT) of a sequence of

n

complex numbers

z_{j}

, for

j = 0, 1, \dots, n - 1

. The direct transform is defined by

{\hat{z}}_{k} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} z_{j} \exp (- i \frac{2 π j k}{n})

(1)

for

k = 0, 1, \dots, n - 1

. Note that equation (1) makes sense for all integral

k

and with this extension

{\hat{z}}_{k}

is periodic with period

n

, i.e.,

{\hat{z}}_{k} = {\hat{z}}_{k \pm n}

, and in particular

{\hat{z}}_{- k} = {\hat{z}}_{n - k}

. Note also that the scale-factor of

\frac{1}{\sqrt{n}}

may be omitted in the definition of the DFT, and replaced by

\frac{1}{n}

in the definition of the inverse.

If we write

z_{j} = x_{j} + i y_{j}

and

{\hat{z}}_{k} = a_{k} + i b_{k}

, then the definition of

{\hat{z}}_{k}

may be written in terms of sines and cosines as

a_{k} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} (x_{j} \cos (\frac{2 π j k}{n}) + y_{j} \sin (\frac{2 π j k}{n}))

b_{k} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} (y_{j} \cos (\frac{2 π j k}{n}) - x_{j} \sin (\frac{2 π j k}{n})) .

The original data values

z_{j}

may conversely be recovered from the transform

{\hat{z}}_{k}

by an inverse discrete Fourier transform:

z_{j} = \frac{1}{\sqrt{n}} \sum_{k = 0}^{n - 1} {\hat{z}}_{k} \exp (+ i \frac{2 π j k}{n})

(2)

for

j = 0, 1, \dots, n - 1

. If we take the complex conjugate of (2), we find that the sequence

{\bar{z}}_{j}

is the DFT of the sequence

{\bar{\hat{z}}}_{k}

. Hence the inverse DFT of the sequence

{\hat{z}}_{k}

may be obtained by taking the complex conjugates of the

{\hat{z}}_{k}

; performing a DFT, and taking the complex conjugates of the result. (Note that the terms forward transform and backward transform are also used to mean the direct and inverse transforms respectively.)

The definition (1) of a one-dimensional transform can easily be extended to multidimensional transforms. For example, in two dimensions we have

{\hat{z}}_{k_{1} k_{2}} = \frac{1}{\sqrt{n_{1} n_{2}}} \sum_{j_{1} = 0}^{n_{1} - 1} \sum_{j_{2} = 0}^{n_{2} - 1} z_{j_{1} j_{2}} \exp (- i \frac{2 π j_{1} k_{1}}{n_{1}}) \exp (- i \frac{2 π j_{2} k_{2}}{n_{2}}) .

(3)

Note: definitions of the discrete Fourier transform vary. Sometimes (2) is used as the definition of the DFT, and (1) as the definition of the inverse.

If the original sequence is purely real valued, i.e.,

z_{j} = x_{j}

, then

{\hat{z}}_{k} = a_{k} + i b_{k} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} x_{j} \exp (- i \frac{2 π j k}{n})

and

{\hat{z}}_{n - k}

is the complex conjugate of

{\hat{z}}_{k}

. Thus the DFT of a real sequence is a particular type of complex sequence, called a Hermitian sequence, or half-complex or conjugate symmetric, with the properties

a_{n - k} = a_{k} b_{n - k} = {- b}_{k} b_{0} = 0

and, if

n

is even,

b_{n / 2} = 0

.

Thus a Hermitian sequence of

n

complex data values can be represented by only

n

, rather than

2 n

, independent real values. This can obviously lead to economies in storage, with two schemes being used in this chapter. In the first (deprecated) scheme, which will be referred to as the real storage format for Hermitian sequences, the real parts

a_{k}

for

0 \leq k \leq n / 2

are stored in normal order in the first

n / 2 + 1

locations of an array x of length

n

; the corresponding nonzero imaginary parts are stored in reverse order in the remaining locations of x. To clarify, if x is declared with bounds

(0 : n - 1)

in your calling subroutine, the following two tables illustrate the storage of the real and imaginary parts of

{\hat{z}}_{k}

for the two cases:

n

even and

n

odd.

If

n

is even then the sequence has two purely real elements and is stored as follows:

Index of x	0	1	2	$\dots$	$n / 2$	$\dots$	$n - 2$	$n - 1$
Sequence	$a_{0}$	$a_{1} + {i b}_{1}$	$a_{2} + {i b}_{2}$	$\dots$	$a_{n / 2}$	$\dots$	$a_{2} - {i b}_{2}$	$a_{1} - {i b}_{1}$
Stored values	$a_{0}$	$a_{1}$	$a_{2}$	$\dots$	$a_{n / 2}$	$\dots$	$b_{2}$	$b_{1}$

\begin{matrix} x (k) = a_{k}, & for ​ k = 0, 1, \dots, n / 2, and \\ x (n - k) = b_{k}, & for ​ k = 1, 2, \dots, n / 2 - 1 . \end{matrix}

If

n

is odd then the sequence has one purely real element and, letting

n = 2 s + 1

, is stored as follows:

Index of x	0	1	2	$\dots$	$s$	$s + 1$	$\dots$	$n - 2$	$n - 1$
Sequence	$a_{0}$	$a_{1} + {i b}_{1}$	$a_{2} + {i b}_{2}$	$\dots$	$a_{s} + i b_{s}$	$a_{s} - i b_{s}$	$\dots$	$a_{2} - {i b}_{2}$	$a_{1} - {i b}_{1}$
Stored values	$a_{0}$	$a_{1}$	$a_{2}$	$\dots$	$a_{s}$	$b_{s}$	$\dots$	$b_{2}$	$b_{1}$

\begin{matrix} x (k) = a_{k}, & for ​ k = 0, 1, \dots, s, and \\ x (n - k) = b_{k}, & for ​ k = 1, 2, \dots, s . \end{matrix}

The second (recommended) storage scheme, referred to in this chapter as the complex storage format for Hermitian sequences, stores the real and imaginary parts

a_{k}, b_{k}

, for

0 \leq k \leq n / 2

, in consecutive locations of an array x of length

n + 2

. If x is declared with bounds

(0 : n + 1)

in your calling subroutine, the following two tables illustrate the storage of the real and imaginary parts of

{\hat{z}}_{k}

for the two cases:

n

even and

n

odd.

If

n

is even then the sequence has two purely real elements and is stored as follows:

Index of x	0	1	2	3	$\dots$	$n - 2$	$n - 1$	$n$	$n + 1$
Stored values	$a_{0}$	$b_{0} = 0$	$a_{1}$	$b_{1}$	$\dots$	$a_{n / 2 - 1}$	$b_{n / 2 - 1}$	$a_{n / 2}$	$b_{n / 2} = 0$

\begin{matrix} x (2 \times k) = a_{k}, & for ​ k = 0, 1, \dots, n / 2, and \\ x (2 \times k + 1) = b_{k}, & for ​ k = 0, 1, \dots, n / 2 . \end{matrix}

If

n

is odd then the sequence has one purely real element and, letting

n = 2 s + 1

, is stored as follows:

Index of x	0	1	2	3	$\dots$	$n - 2$	$n - 1$	$n$	$n + 1$
Stored values	$a_{0}$	$b_{0} = 0$	$a_{1}$	$b_{1}$	$\dots$	$b_{s - 1}$	$a_{s}$	$b_{s}$	$0$

\begin{matrix} x (2 \times k) = a_{k}, & for ​ k = 0, 1, \dots, s, and \\ x (2 \times k + 1) = b_{k}, & for ​ k = 0, 1, \dots, s . \end{matrix}

Also, given a Hermitian sequence, the inverse (or backward) discrete transform produces a real sequence. That is,

x_{j} = \frac{1}{\sqrt{n}} (a_{0} + 2 \sum_{k = 1}^{n / 2 - 1} (a_{k} \cos (\frac{2 π j k}{n}) - b_{k} \sin (\frac{2 π j k}{n})) + a_{n / 2})

where

a_{n / 2} = 0

if

n

is odd.

For real data that is two-dimensional or higher, the symmetry in the transform persists for the leading dimension only. So, using the notation of equation (3) for the complex two-dimensional discrete transform, we have that

{\hat{z}}_{k_{1} k_{2}}

is the complex conjugate of

{\hat{z}}_{(n_{1} - k_{1}) (n_{2} - k_{2})}

. It is more convenient for transformed data of two or more dimensions to be stored as a complex sequence of length

(n_{1} / 2 + 1) \times n_{2} \times \dots \times n_{d}

where

d

is the number of dimensions. The inverse discrete Fourier transform operating on such a complex sequence (Hermitian in the leading dimension) returns a real array of full dimension (

n_{1} \times n_{2} \times \dots \times n_{d}

).

In many applications the sequence

x_{j}

will not only be real, but may also possess additional symmetries which we may exploit to reduce further the computing time and storage requirements. For example, if the sequence

x_{j}

is odd,

(x_{j} = {- x}_{n - j})

, then the discrete Fourier transform of

x_{j}

contains only sine terms. Rather than compute the transform of an odd sequence, we define the sine transform of a real sequence by

{\hat{x}}_{k} = \sqrt{\frac{2}{n}} \sum_{j = 1}^{n - 1} x_{j} \sin (\frac{π j k}{n}),

which could have been computed using the Fourier transform of a real odd sequence of length

2 n

. In this case the

x_{j}

are arbitrary, and the symmetry only becomes apparent when the sequence is extended. Similarly we define the cosine transform of a real sequence by

{\hat{x}}_{k} = \sqrt{\frac{2}{n}} (\frac{1}{2} x_{0} + \sum_{j = 1}^{n - 1} x_{j} \cos (\frac{π j k}{n}) + \frac{1}{2} {(- 1)}^{k} x_{n})

which could have been computed using the Fourier transform of a real even sequence of length

2 n

.

In addition to these ‘half-wave’ symmetries described above, sequences arise in practice with ‘quarter-wave’ symmetries. We define the quarter-wave sine transform by

{\hat{x}}_{k} = \frac{1}{\sqrt{n}} (\sum_{j = 1}^{n - 1} x_{j} \sin (\frac{π j (2 k - 1)}{2 n}) + \frac{1}{2} {(- 1)}^{k - 1} x_{n})

which could have been computed using the Fourier transform of a real sequence of length

4 n

of the form

(0, x_{1}, \dots, x_{n}, x_{n - 1}, \dots, x_{1}, 0, {- x}_{1}, \dots, {- x}_{n}, {- x}_{n - 1}, \dots, {- x}_{1}) .

Similarly we may define the quarter-wave cosine transform by

{\hat{x}}_{k} = \frac{1}{\sqrt{n}} (\frac{1}{2} x_{0} + \sum_{j = 1}^{n - 1} x_{j} \cos (\frac{π j (2 k - 1)}{2 n}))

which could have been computed using the Fourier transform of a real sequence of length

4 n

of the form

(x_{0}, x_{1}, \dots, x_{n - 1}, 0, {- x}_{n - 1}, \dots, {- x}_{0}, {- x}_{1}, \dots, {- x}_{n - 1}, 0, x_{n - 1}, \dots, x_{1}) .

The usual application of the discrete Fourier transform is that of obtaining an approximation of the Fourier integral transform

F (s) = \int_{- \infty}^{\infty} f (t) \exp (- i 2 π s t) d t

when

f (t)

is negligible outside some region

(0, c)

. Dividing the region into

n

equal intervals we have

F (s) ≅ \frac{c}{n} \sum_{j = 0}^{n - 1} f_{j} \exp (\frac{- i 2 π s j c}{n})

and so

F_{k} ≅ \frac{c}{n} \sum_{j = 0}^{n - 1} f_{j} \exp (\frac{- i 2 π j k}{n})

for

k = 0, 1, \dots, n - 1

, where

f_{j} = f (j c / n)

and

F_{k} = F (k / c)

.

Hence the discrete Fourier transform gives an approximation to the Fourier integral transform in the region

s = 0

to

s = n / c

.

If the function

f (t)

is defined over some more general interval

(a, b)

, then the integral transform can still be approximated by the discrete transform provided a shift is applied to move the point

a

to the origin.

One of the most important applications of the discrete Fourier transform is to the computation of the discrete convolution or correlation of two vectors

x

and

y

defined (as in Brigham (1974)) by

convolution: $z_{k} = \sum_{j = 0}^{n - 1} x_{j} y_{k - j}$
correlation: $w_{k} = \sum_{j = 0}^{n - 1} {\bar{x}}_{j} y_{k + j}$

(Here

x

and

y

are assumed to be periodic with period

n

.)

Under certain circumstances (see Brigham (1974)) these can be used as approximations to the convolution or correlation integrals defined by

z (s) = \int_{- \infty}^{\infty} x (t) y (s - t) d t

and

w (s) = \int_{- \infty}^{\infty} \bar{x} (t) y (s + t) d t, - \infty < s < \infty .

For more general advice on the use of Fourier transforms, see Hamming (1962); more detailed information on the fast Fourier transform algorithm can be found in Gentleman and Sande (1966) and Brigham (1974).

A further application of the fast Fourier transform, and in particular of the Fourier transforms of symmetric sequences, is in the solution of elliptic PDEs. If an equation is discretized using finite differences, then it is possible to reduce the problem of solving the resulting large system of linear equations to that of solving a number of tridiagonal systems of linear equations. This is accomplished by uncoupling the equations using Fourier transforms, where the nature of the boundary conditions determines the choice of transforms – see Section 3.3. Full details of the Fourier method for the solution of PDEs may be found in Swarztrauber (1977) and Swarztrauber (1984).

Let

f (t)

be a real function of

t

, with

f (t) = 0

for

t < 0

, and be piecewise continuous and of exponential order

α

, i.e.,

|f (t)| \leq M e^{α t}

for large

t

, where

α

is the minimal such exponent.

The Laplace transform of

f (t)

is given by

F (s) = \int_{0}^{\infty} e^{- s t} f (t) d t, t > 0

where

F (s)

is defined for

Re (s) > α

.

The inverse transform is defined by the Bromwich integral

f (t) = \frac{1}{2 π i} \int_{a - i \infty}^{a + i \infty} e^{s t} F (s) d s, t > 0 .

The integration is performed along the line

s = a

in the complex plane, where

a > α

. This is equivalent to saying that the line

s = a

lies to the right of all singularities of

F (s)

. For this reason, the value of

α

is crucial to the correct evaluation of the inverse. It is not essential to know

α

exactly, but an upper bound must be known.

The problem of determining an inverse Laplace transform may be classified according to whether (a)

F (s)

is known for real values only, or (b)

F (s)

is known in functional form and can therefore be calculated for complex values of

s

. Problem (a) is very ill-defined and no routines are provided. Two methods are provided for problem (b).

Gauss transforms have applications in areas including statistics, machine learning, and numerical solution of the heat equation. The discrete Gauss transform (DGT),

G (y)

, evaluated at a set of target points

y (j)

, for

j = 1, 2, \dots, m \in ℝ^{d}

, is defined as:

G (y_{j}) = \sum_{i = 1}^{n} q_{i} e^{- {‖y_{j} - x_{i}‖}_{2}^{2} / h_{i}^{2}}, j = 1, \dots, m

where

x_{i}

, for

i = 1, 2, \dots, n \in ℝ^{d}

, are the Gaussian source points,

q_{i}

, for

i = 1, 2, \dots, n \in ℝ^{+}

, are the source weights and

h_{i}

, for

i = 1, 2, \dots, n \in ℝ^{+}

, are the source standard deviations (alternatively source scales or source bandwidths).

The fast Gauss transform (FGT) algorithm presented in Raykar and Duraiswami (2005) approximates the DGT by using two Taylor series and clustering of the source points.

For any series of functions

ϕ_{i}

which satisfy a recurrence

ϕ_{r + 1} (x) + α_{r} (x) ϕ_{r} (x) + β_{r} (x) ϕ_{r - 1} (x) = 0

the sum

\sum_{r = 0}^{n} a_{r} ϕ_{r} (x)

is given by

\sum_{r = 0}^{n} a_{r} ϕ_{r} (x) = b_{0} (x) ϕ_{0} (x) + b_{1} (x) (ϕ_{1} (x) + α_{0} (x) ϕ_{0} (x))

where

b_{r} (x) + α_{r} (x) b_{r + 1} (x) + β_{r + 1} (x) b_{r + 2} (x) = a_{r} b_{n + 1} (x) = b_{n + 2} (x) = 0 .

This may be used to compute the sum of the series. For further reading, see Hamming (1962).

This device has applications in a large number of fields, such as summation of series, calculation of integrals with oscillatory integrands (including, for example, Hankel transforms), and root-finding. The mathematical description is as follows. Given a sequence of values

\{s_{n}\}

, for

n = m, \dots, m + 2 l

, then, except in certain singular cases, arguments,

a

,

b_{i}

,

c_{i}

may be determined such that

s_{n} = a + \sum_{i = 1}^{l} b_{i} c_{i}^{n} .

If the sequence

\{s_{n}\}

converges, then

a

may be taken as an estimate of the limit. The method will also find a pseudo-limit of certain divergent sequences – see Shanks (1955) for details.

To use the method to sum a series, the terms

s_{n}

of the sequence should be the partial sums of the series, e.g.,

s_{n} = \sum_{k = 1}^{n} t_{k}

, where

t_{k}

is the

k

th term of the series. The algorithm can also be used to some advantage to evaluate integrals with oscillatory integrands; one approach is to write the integral (in this case over a semi-infinite interval) as

\int_{0}^{\infty} f (x) d x = \int_{0}^{a_{1}} f (x) d x + \int_{a_{1}}^{a_{2}} f (x) d x + \int_{a_{2}}^{a_{3}} f (x) d x + \dots

and to consider the sequence of values

s_{1} = \int_{0}^{a_{1}} f (x) d x, s_{2} = \int_{0}^{a_{2}} f (x) d x = s_{1} + \int_{a_{1}}^{a_{2}} f (x) d x, etc.,

where the integrals are evaluated using standard quadrature methods. In choosing the values of the

a_{k}

, it is worth bearing in mind that c06baf converges much more rapidly for sequences whose values oscillate about a limit. The

a_{k}

should thus be chosen to be (close to) the zeros of

f (x)

, so that successive contributions to the integral are of opposite sign. As an example, consider the case where

f (x) = M (x) \sin x

and

M (x) > 0

: convergence will be much improved if

a_{k} = k π

rather than

a_{k} = 2 k π

.

The fast Fourier transform algorithm ceases to be ‘fast’ if applied to values of

n

which cannot be expressed as a product of small prime factors. All the FFT routines in this chapter are particularly efficient if the only prime factors of

n

are

2

,

3

or

5

.

The choice of routine is determined first of all by whether the data values constitute a real, Hermitian or general complex sequence. It is wasteful of time and storage to use an inappropriate routine.

c06paf transforms a single sequence of real data onto (and in-place) a representation of the transformed Hermitian sequence using the complex storage scheme described in Section 2.1.2. c06paf also performs the inverse transform using the representation of Hermitian data and transforming back to a real data sequence.

Alternatively, the two-dimensional routine c06pvf can be used (on setting the second dimension to 1) to transform a sequence of real data onto an Hermitian sequence whose first half is stored in a separate Complex array. The second half need not be stored since these are the complex conjugate of the first half in reverse order. c06pwf performs the inverse operation, transforming the the Hermitian sequence (half-)stored in a Complex array onto a separate real array.

Multiple sequences of real data can also be transformed to and from a transformed Hermitian sequence (using the complex storage scheme). c06ppf assumes that the original real data is stored such that all the corresponding (

j

-th say) elements of the different streams are stored contiguously. c06pqf assumes that the elements of each sequence are stored contiguously; this is the recommended form of storage since it normally results in more efficient computation.

c06pcf transforms a single complex sequence in-place; it also performs the inverse transform. c06psf transforms multiple complex sequences, each stored sequentially; it also performs the inverse transform on multiple complex sequences. This routine is designed to perform several transforms in a single call, all with the same value of

n

.

If extensive use is to be made of these routines and you are concerned about efficiency, you are advised to conduct your own timing tests.

Four routines are provided for computing fast Fourier transforms (FFTs) of real symmetric sequences. c06ref computes multiple Fourier sine transforms, c06rff computes multiple Fourier cosine transforms, c06rgf computes multiple quarter-wave Fourier sine transforms, and c06rhf computes multiple quarter-wave Fourier cosine transforms.

As described in Section 2.1.6, Fourier transforms may be used in the solution of elliptic PDEs.

c06ref may be used to solve equations where the solution is specified along the boundary.

c06rff may be used to solve equations where the derivative of the solution is specified along the boundary.

c06rgf may be used to solve equations where the solution is specified on the lower boundary, and the derivative of the solution is specified on the upper boundary.

c06rhf may be used to solve equations where the derivative of the solution is specified on the lower boundary, and the solution is specified on the upper boundary.

For equations with periodic boundary conditions the full-range Fourier transforms computed by c06paf are appropriate.

The following routines compute multidimensional discrete Fourier transforms of real, Hermitian and complex data stored in complex arrays:

	real	Hermitian	complex
2 dimensions	c06pvf	c06pwf	c06puf
3 dimensions	c06pyf	c06pzf	c06pxf
any number of dimensions			c06pjf

The Hermitian data, either transformed from or being transformed to real data, is compacted (due to symmetry) along its first dimension when stored in Complex arrays; thus approximately half the full Hermitian data is stored.

c06puf and c06pxf should be used in preference to c06pjf for two- and three-dimensional transforms, as they are easier to use and are likely to be more efficient.

The transform of multidimensional real data is stored as a complex sequence that is Hermitian in its leading dimension. The inverse transform takes such a complex sequence and computes the real transformed sequence. Consequently, separate routines are provided for performing forward and inverse transforms.

c06pvf performs the forward two-dimensionsal transform while c06pwf performs the inverse of this transform.

c06pyf performs the forward three-dimensional transform while c06pzf performs the inverse of this transform.

The complex sequences computed by c06pvf and c06pyf contain roughly half of the Fourier coefficients; the remainder can be reconstructed by conjugation of those computed. For example, the Fourier coefficients of the two-dimensional transform

{\hat{z}}_{(n_{1} - k_{1}) k_{2}}

are the complex conjugate of

{\hat{z}}_{k_{1} k_{2}}

for

k_{1} = 0, 1, \dots, n_{1} / 2

, and

k_{2} = 0, 1, \dots, n_{2} - 1

.

c06fkf computes either the discrete convolution or the discrete correlation of two real vectors.

c06pkf computes either the discrete convolution or the discrete correlation of two complex vectors.

Two methods are provided: Weeks' method (c06lbf) and Crump's method (c06laf). Both require the function

F (s)

to be evaluated for complex values of

s

. If in doubt which method to use, try Weeks' method (c06lbf) first; when it is suitable, it is usually much faster.

Typically the inversion of a Laplace transform becomes harder as

t

increases so that all numerical methods tend to have a limit on the range of

t

for which the inverse

f (t)

can be computed. c06laf is useful for small and moderate values of

t

.

It is often convenient or necessary to scale a problem so that

α

is close to

0

. For this purpose it is useful to remember that the inverse of

F (s + k)

is

\exp (- k t) f (t)

. The method used by c06laf is not so satisfactory when

f (t)

is close to zero, in which case a term may be added to

F (s)

, e.g.,

k / s + F (s)

has the inverse

k + f (t)

.

Singularities in the inverse function

f (t)

generally cause numerical methods to perform less well. The positions of singularities can often be identified by examination of

F (s)

. If

F (s)

contains a term of the form

\exp (- k s) / s

then a finite discontinuity may be expected in the inverse at

t = k

. c06laf, for example, is capable of estimating a discontinuous inverse but, as the approximation used is continuous, Gibbs' phenomena (overshoots around the discontinuity) result. If possible, such singularities of

F (s)

should be removed before computing the inverse.

The only routine available is c06saf. If the dimensionality of the data is low or the number of source and target points is small, however, it may be more efficient to evaluate the discrete Gauss transform directly.

The only routine available is c06dcf, which sums a finite Chebyshev series

\sum_{j = 0}^{n} c_{j} T_{j} (x), \sum_{j = 0}^{n} c_{j} T_{2 j} (x) or \sum_{j = 0}^{n} c_{j} T_{2 j + 1} (x)

depending on the choice of argument.

The only routine available is c06baf.

None.

The following lists all those routines that have been withdrawn since Mark 23 of the Library or are in the Library, but deprecated.

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall

Davies S B and Martin B (1979) Numerical inversion of the Laplace transform: A survey and comparison of methods J. Comput. Phys. 33 1–32

Fox L and Parker I B (1968) Chebyshev Polynomials in Numerical Analysis Oxford University Press

Gentleman W S and Sande G (1966) Fast Fourier transforms for fun and profit Proc. Joint Computer Conference, AFIPS 29 563–578

Hamming R W (1962) Numerical Methods for Scientists and Engineers McGraw–Hill

Raykar V C and Duraiswami R (2005) Improved Fast Gauss Transform With Variable Source Scales University of Maryland Technical Report CS-TR-4727/UMIACS-TR-2005-34

Shanks D (1955) Nonlinear transformations of divergent and slowly convergent sequences J. Math. Phys. 34 1–42

Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19(3) 490–501

Swarztrauber P N (1984) Fast Poisson solvers Studies in Numerical Analysis (ed G H Golub) Mathematical Association of America

Swarztrauber P N (1986) Symmetric FFT's Math. Comput. 47(175) 323–346

Wynn P (1956) On a device for computing the

e_{m} (S_{n})

transformation Math. Tables Aids Comput. 10 91–96

Is the data one-dimensional?			Multiple vectors?			Stored as rows?			c06prf
		yes			yes			yes
	no			no			no
						Stored as columns?			c06psf
								yes

			c06pcf


Is the data two-dimensional?			c06puf
		yes
	no
Is the data three-dimensional?			c06pxf
		yes
	no
Transform on one dimension only?			c06pff
		yes
	no
Transform on all dimensions?			c06pjf
		yes

Quarter-wave sine (inverse) transform?			c06rgf
		yes
	no
Quarter-wave cosine (inverse) transform?			c06rhf
		yes
	no
Sine (inverse) transform?			c06ref
		yes
	no
Cosine (inverse) transform?			c06rff
		yes
	no
Is the data three-dimensional?			Forward transform on real data?			c06pyf
		yes			yes
	no			no
			Inverse transform on Hermitian data?			c06pzf
					yes

Is the data two-dimensional?			Forward transform on real data?			c06pvf
		yes			yes
	no			no
			Inverse transform on Hermitian data?			c06pwf
					yes

Is the data multi one-dimensional?			Sequence elements stored by stride (number of sequences)?			c06ppf
		yes			yes
	no			no
			Sequence elements stored contiguously?			c06pqf
					yes

c06paf

Routine	Status	Replacement Routine(s)
c06dbf	Withdrawn at Mark 25	c06dcf
c06eaf	Withdrawn at Mark 26	c06paf
c06ebf	Withdrawn at Mark 26	c06paf
c06ecf	Withdrawn at Mark 26	c06pcf
c06ekf	Withdrawn at Mark 26	c06fkf
c06fpf	To be withdrawn at Mark 28	c06ppf or c06pqf
c06fqf	To be withdrawn at Mark 28	c06ppf or c06pqf
c06frf	Withdrawn at Mark 26	c06psf
c06fuf	Withdrawn at Mark 26	c06puf
c06gbf	Withdrawn at Mark 26	No replacement required
c06gcf	Withdrawn at Mark 26	No replacement required
c06gqf	Withdrawn at Mark 26	No replacement required
c06gsf	Withdrawn at Mark 26	No replacement required
c06haf	Withdrawn at Mark 26	c06ref
c06hbf	Withdrawn at Mark 26	c06rff
c06hcf	Withdrawn at Mark 26	c06rgf
c06hdf	Withdrawn at Mark 26	c06rhf

NAG FL Interface
C06 (Sum)
Summation of Series

▸▿ Contents

1 Scope of the Chapter

2 Background to the Problems

2.1 Discrete Fourier Transforms

2.1.1 Complex transforms

2.1.2 Real transforms

2.1.3 Real symmetric transforms

2.1.4 Fourier integral transforms

2.1.5 Convolutions and correlations

2.1.6 Applications to solving partial differential equations (PDEs)

2.2 Inverse Laplace Transforms

2.3 Fast Gauss Transform

2.4 Direct Summation of Orthogonal Series

2.5 Acceleration of Convergence

3 Recommendations on Choice and Use of Available Routines

3.1 One-dimensional Fourier Transforms

3.1.1 Real and Hermitian data

3.1.2 Complex data

3.2 Half- and Quarter-wave Transforms

3.3 Application to Elliptic Partial Differential Equations

3.4 Multidimensional Fourier Transforms

3.5 Convolution and Correlation

3.6 Inverse Laplace Transforms

3.7 Fast Gauss Transform

3.8 Direct Summation of Orthogonal Series

3.9 Acceleration of Convergence

4 Decision Trees

Tree 1: Fourier Transform of Discrete Complex Data

Tree 2: Fourier Transform of Real Data or Data in Complex Hermitian Form Resulting from the Transform of Real Data

5 Functionality Index

6 Auxiliary Routines Associated with Library Routine Arguments

7 Withdrawn or Deprecated Routines

8 References

NAG FL InterfaceC06 (Sum)Summation of Series

▸▿ Contents

1 Scope of the Chapter

2 Background to the Problems

2.1 Discrete Fourier Transforms

2.1.1 Complex transforms

2.1.2 Real transforms

2.1.3 Real symmetric transforms

2.1.4 Fourier integral transforms

2.1.5 Convolutions and correlations

2.1.6 Applications to solving partial differential equations (PDEs)

2.2 Inverse Laplace Transforms

2.3 Fast Gauss Transform

2.4 Direct Summation of Orthogonal Series

2.5 Acceleration of Convergence

3 Recommendations on Choice and Use of Available Routines

3.1 One-dimensional Fourier Transforms

3.1.1 Real and Hermitian data

3.1.2 Complex data

3.2 Half- and Quarter-wave Transforms

3.3 Application to Elliptic Partial Differential Equations

3.4 Multidimensional Fourier Transforms

3.5 Convolution and Correlation

3.6 Inverse Laplace Transforms

3.7 Fast Gauss Transform

3.8 Direct Summation of Orthogonal Series

3.9 Acceleration of Convergence

4 Decision Trees

Tree 1: Fourier Transform of Discrete Complex Data

Tree 2: Fourier Transform of Real Data or Data in Complex Hermitian Form Resulting from the Transform of Real Data

5 Functionality Index

6 Auxiliary Routines Associated with Library Routine Arguments

7 Withdrawn or Deprecated Routines

8 References

NAG FL Interface
C06 (Sum)
Summation of Series