C06 Chapter Contents
NAG Library Manual

NAG Library Chapter IntroductionC06 – Summation of Series

1  Scope of the Chapter

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 function. (d) Direct summation of orthogonal series. (e) Acceleration of convergence of a sequence of real values.

2  Background to the Problems

2.1  Discrete Fourier Transforms

2.1.1  Complex transforms

Most of the routines in this chapter calculate the finite discrete Fourier transform (DFT) of a sequence of $n$ complex numbers ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$. The direct transform is defined by
 $z^k = 1n ∑ j=0 n-1 zj exp -i 2πjk n$ (1)
for $k=0,1,\dots ,n-1$. Note that equation (1) makes sense for all integral $k$ and with this extension ${\stackrel{^}{z}}_{k}$ is periodic with period $n$, i.e., ${\stackrel{^}{z}}_{k}={\stackrel{^}{z}}_{k±n}$, and in particular ${\stackrel{^}{z}}_{-k}={\stackrel{^}{z}}_{n-k}$. 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 ${\stackrel{^}{z}}_{k}={a}_{k}+i{b}_{k}$, then the definition of ${\stackrel{^}{z}}_{k}$ may be written in terms of sines and cosines as
 $ak = 1n ∑ j=0 n-1 xj cos 2πjk n + yj sin 2πjk n$
 $bk = 1n ∑ j= 0 n- 1 yj cos 2πjk n - xj sin 2πjk n .$
The original data values ${z}_{j}$ may conversely be recovered from the transform ${\stackrel{^}{z}}_{k}$ by an inverse discrete Fourier transform:
 $zj = 1n ∑ k=0 n-1 z^k exp +i 2πjk n$ (2)
for $j=0,1,\dots ,n-1$. If we take the complex conjugate of (2), we find that the sequence ${\stackrel{-}{z}}_{j}$ is the DFT of the sequence ${\stackrel{-}{\stackrel{^}{z}}}_{k}$. Hence the inverse DFT of the sequence ${\stackrel{^}{z}}_{k}$ may be obtained by taking the complex conjugates of the ${\stackrel{^}{z}}_{k}$; 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
 $z^ k1 k2 = 1 n1 n2 ∑ j1=0 n1-1 ∑ j2=0 n2-1 z j1 j2 exp -i 2 π j1 k1 n1 exp -i 2 π j2 k2 n2 .$ (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.

2.1.2  Real transforms

If the original sequence is purely real valued, i.e., ${z}_{j}={x}_{j}$, then
 $z^k = ak + i bk = 1n ∑ j=0 n-1 xj exp -i 2πjk n$
and ${\stackrel{^}{z}}_{n-k}$ is the complex conjugate of ${\stackrel{^}{z}}_{k}$. 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
 $a n-k = ak b n-k = -bk b0 = 0$
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 $2n$, independent real values. This can obviously lead to economies in storage, with two schemes being used in this chapter. In the first scheme, which will be referred to as the real storage format for Hermitian sequences, the real parts ${a}_{k}$ for $0\le k\le n/2$ 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 $\left(0:n-1\right)$ in your calling subroutine, the following two tables illustrate the storage of the real and imaginary parts of ${\stackrel{^}{z}}_{k}$ 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 ${a}_{0}$ ${a}_{1}+{ib}_{1}$ ${a}_{2}+{ib}_{2}$ $\dots$ ${a}_{n/2}$ $\dots$ ${a}_{2}-{ib}_{2}$ ${a}_{1}-{ib}_{1}$ Stored values ${a}_{0}$ ${a}_{1}$ ${a}_{2}$ $\dots$ ${a}_{n/2}$ $\dots$ ${b}_{2}$ ${b}_{1}$
 $Xk = ak , for ​ k= 0, 1, …, n/2 , and Xn-k = bk , for ​ k= 1, 2, …, n/2-1 .$
If $n$ is odd then the sequence has one purely real element and, letting $n=2s+1$, is stored as follows:
 Index of X 0 1 2 $\dots$ $s$ $s+1$ $\dots$ $n-2$ $n-1$ Sequence ${a}_{0}$ ${a}_{1}+{ib}_{1}$ ${a}_{2}+{ib}_{2}$ $\dots$ ${a}_{s}+i{b}_{s}$ ${a}_{s}-i{b}_{s}$ $\dots$ ${a}_{2}-{ib}_{2}$ ${a}_{1}-{ib}_{1}$ Stored values ${a}_{0}$ ${a}_{1}$ ${a}_{2}$ $\dots$ ${a}_{s}$ ${b}_{s}$ $\dots$ ${b}_{2}$ ${b}_{1}$
 $Xk = ak , for ​ k= 0, 1, …, s , and Xn-k = bk , for ​ k= 1, 2, …, s .$
The second 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 $0\le k\le n/2$, in consecutive locations of an array X of length $n+2$. If X is declared with bounds $\left(0:n+1\right)$ in your calling subroutine, the following two tables illustrate the storage of the real and imaginary parts of ${\stackrel{^}{z}}_{k}$ 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 ${a}_{0}$ ${b}_{0}=0$ ${a}_{1}$ ${b}_{1}$ $\dots$ ${a}_{n/2-1}$ ${b}_{n/2-1}$ ${a}_{n/2}$ ${b}_{n/2}=0$
 $X2×k = ak , for ​ k= 0, 1, …, n/2 , and X2×k+1 = bk , for ​ k= 0, 1, …, n/2 .$
If $n$ is odd then the sequence has one purely real element and, letting $n=2s+1$, is stored as follows:
 Index of X 0 1 2 3 $\dots$ $n-2$ $n-1$ $n$ $n+1$ Stored values ${a}_{0}$ ${b}_{0}=0$ ${a}_{1}$ ${b}_{1}$ $\dots$ ${b}_{s-1}$ ${a}_{s}$ ${b}_{s}$ $0$
 $X2×k = ak , for ​ k= 0, 1, …, s , and X2×k+1 = bk , for ​ k= 0, 1, …, s .$
Also, given a Hermitian sequence, the inverse (or backward) discrete transform produces a real sequence. That is,
 $xj = 1n a0 + 2 ∑ k=1 n/2-1 ak cos 2πjk n - bk sin 2πjk n + a n/2$
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 ${\stackrel{^}{z}}_{{k}_{1}{k}_{2}}$ is the complex conjugate of ${\stackrel{^}{z}}_{\left({n}_{1}-{k}_{1}\right){k}_{2}}$. It is more convenient for transformed data of two or more dimensions to be stored as a complex sequence of length $\left({n}_{1}/2+1\right)×{n}_{2}×\dots ×{n}_{d}$ 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 (${n}_{1}×{n}_{2}×\dots ×{n}_{d}$).

2.1.3  Real symmetric transforms

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, $\left({x}_{j}={-x}_{n-j}\right)$, 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
 $x^k = 2n ∑j=1 n-1 xj sin πjkn ,$
which could have been computed using the Fourier transform of a real odd sequence of length $2n$. 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
 $x^k = 2n 12 x0 + ∑ j=1 n-1 xj cos πjkn + 12 -1k xn$
which could have been computed using the Fourier transform of a real even sequence of length $2n$.
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
 $x^k = 1n ∑ j=1 n-1 xj sin π j 2k-1 2n + 12 -1 k-1 xn$
which could have been computed using the Fourier transform of a real sequence of length $4n$ of the form
 $0,x1,…,xn,xn-1 ,…,x1,0,-x1,…,-xn, -x n-1 ,…, -x 1 .$
Similarly we may define the quarter-wave cosine transform by
 $x^k = 1n 12 x0 + ∑j= 1 n- 1 xj cos π j 2k- 1 2n$
which could have been computed using the Fourier transform of a real sequence of length $4n$ of the form
 $x0,x1,…, x n-1 ,0, -x n-1 ,…,-x0,-x1,…, -x n-1 ,0, x n-1 ,…,x1 .$

2.1.4  Fourier integral transforms

The usual application of the discrete Fourier transform is that of obtaining an approximation of the Fourier integral transform
 $F s = ∫ -∞ ∞ f t exp -i 2 π s t dt$
when $f\left(t\right)$ is negligible outside some region $\left(0,c\right)$. Dividing the region into $n$ equal intervals we have
 $F s ≅ cn ∑ j=0 n-1 fj exp -i 2 π s j c n$
and so
 $Fk ≅ cn ∑ j= 0 n- 1 fj exp -i 2 π jk n$
for $k=0,1,\dots ,n-1$, where ${f}_{j}=f\left(jc/n\right)$ and ${F}_{k}=F\left(k/c\right)$.
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 $f\left(t\right)$ is defined over some more general interval $\left(a,b\right)$, 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.

2.1.5  Convolutions and correlations

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: ${w}_{k}=\sum _{j=0}^{n-1}{\stackrel{-}{x}}_{j}{y}_{k+j}$
(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
 $z s = ∫ -∞ ∞ x t y s-t dt$
and
 $w s = ∫ -∞ ∞ x- t y s+t dt , -∞ < s < ∞ .$
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).

2.1.6  Applications to solving partial differential equations (PDEs)

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).

2.2  Inverse Laplace Transforms

Let $f\left(t\right)$ be a real function of $t$, with $f\left(t\right)=0$ for $t<0$, and be piecewise continuous and of exponential order $\alpha$, i.e.,
 $ft ≤ M eαt$
for large $t$, where $\alpha$ is the minimal such exponent.
The Laplace transform of $f\left(t\right)$ is given by
 $F s = ∫0∞ e-st f t dt , t>0$
where $F\left(s\right)$ is defined for $\mathrm{Re}\left(s\right)>\alpha$.
The inverse transform is defined by the Bromwich integral
 $f t = 12πi ∫ a-i∞ a+i∞ est F s ds , t>0 .$
The integration is performed along the line $s=a$ in the complex plane, where $a>\alpha$. This is equivalent to saying that the line $s=a$ lies to the right of all singularities of $F\left(s\right)$. For this reason, the value of $\alpha$ is crucial to the correct evaluation of the inverse. It is not essential to know $\alpha$ exactly, but an upper bound must be known.
The problem of determining an inverse Laplace transform may be classified according to whether (a) $F\left(s\right)$ is known for real values only, or (b) $F\left(s\right)$ 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).

2.3  Direct Summation of Orthogonal Series

For any series of functions ${\varphi }_{i}$ which satisfy a recurrence
 $ϕr+1 x + αr x ϕr x + βr x ϕr-1 x =0$
the sum
 $∑ r= 0 n ar ϕr x$
is given by
 $∑ r=0 n ar ϕr x = b0 x ϕ0 x + b1 x ϕ1 x + α0 x ϕ0 x$
where
 $br x + αr x br+ 1 x + βr+ 1 x br+ 2 x = ar b n+ 1 x = b n+ 2 x = 0 .$
This may be used to compute the sum of the series. For further reading, see Hamming (1962).

2.4  Acceleration of Convergence

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 $\left\{{s}_{n}\right\}$, for $\mathit{n}=m,\dots ,m+2l$, then, except in certain singular cases, parameters, $a$, ${b}_{i}$, ${c}_{i}$ may be determined such that
 $sn = a + ∑ i=1 l bi cin .$
If the sequence $\left\{{s}_{n}\right\}$ 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
 $∫0∞ f x dx = ∫0a1 f x dx + ∫ a1 a2 f x dx + ∫ a2 a3 f x dx + …$
and to consider the sequence of values
 $s1 = ∫ 0 a1 f x dx , s2 = ∫ 0 a2 f x dx = s1 + ∫ a1 a2 f x dx , etc.,$
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 $f\left(x\right)$, so that successive contributions to the integral are of opposite sign. As an example, consider the case where $f\left(x\right)=M\left(x\right)\mathrm{sin}x$ and $M\left(x\right)>0$: convergence will be much improved if ${a}_{k}=k\pi$ rather than ${a}_{k}=2k\pi$.

3  Recommendations on Choice and Use of Available Routines

3.1  One-dimensional Fourier Transforms

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.
The choice is next determined by how you prefer to store complex data. Real storage format routines store general complex sequences and Hermitian sequences in real arrays. In the case of general complex sequences, the real and imaginary parts are stored separately in two real arrays. Since Hermitian sequences contain some symmetries, these can be stored in a compact form in a single real array. Alternatively, complex storage format routines store the corresponding sequence as a complex array for general sequences, and with real and imaginary parts in contiguous locations of a real array for Hermitian sequences.
Two groups, each of three routines, are provided in real storage format and three groups of two routines are provided in complex storage format.
 Group 1 Group 2 Group 3 Real storage format Real sequences C06FAF C06FPF Hermitian sequences C06FBF C06FQF General complex sequences C06FCF Complex storage format/ complex data type Real/Hermitian sequences C06PAF C06PPF C06PQF General complex sequences C06PCF C06PRF C06PSF
Group 1 routines compute a single transform of length $n$, requiring one additional workspace array. Some of these (C06FAF, C06FBF and C06FCF) impose some restrictions on the value of $n$, namely that no prime factor of $n$ may exceed $19$ and the total number of prime factors (including repetitions) may not exceed $20$ (though the latter restriction only becomes relevant when $n>{10}^{6}$).
Group 2 and Group 3 routines are all designed to perform several transforms in a single call, all with the same value of $n$. They are likely to be much faster than the Group 1 routines on modern acrchitectures. They do however require more working storage. It is therefore recommended that, for real storage format, Group 2 routines be used in preference to Group 1 routines, even when only one transform is to be performed. Group 2 and Group 3 routines differ in the way sequences are stored: Group 2 routines store sequences as rows of a two-dimensional array while Group 3 routines store sequences as columns of a two-dimensional array. Group 2 and Group 3 routines impose no practical restrictions on the value of $n$; however, 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 above routines are particularly efficient if the only prime factors of $n$ are $2$, $3$ or $5$.
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.
To compute inverse (backward) discrete Fourier transforms the real storage format routines should be used in conjunction with the complex conjugate of a Hermitian or general sequence of complex data values. In the case of complex storage format routines, there is a direction parameter which determines the direction of the transform; a call to such a routine in the forward direction followed by a call in the backward direction reproduces the original data.

3.2  Half- and Quarter-wave Transforms

Four routines are provided for computing fast Fourier transforms (FFTs) of real symmetric sequences. C06RAF computes multiple Fourier sine transforms, C06RBF computes multiple Fourier cosine transforms, C06RCF computes multiple quarter-wave Fourier sine transforms, and C06RDF computes multiple quarter-wave Fourier cosine transforms.

3.3  Application to Elliptic Partial Differential Equations

As described in Section 2.1, Fourier transforms may be used in the solution of elliptic PDEs.
C06RAF may be used to solve equations where the solution is specified along the boundary.
C06RBF may be used to solve equations where the derivative of the solution is specified along the boundary.
C06RCF 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.
C06RDF 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 C06FPF and C06FQF are appropriate.

3.4  Multidimensional Fourier Transforms

3.4.1  Complex data

The following routines compute multidimensional discrete Fourier transforms of complex data:
 Real storage Complex storage 2 dimensions C06PUF 3 dimensions C06FXF C06PXF any number of dimensions C06FJF C06PJF
The real storage format routines store sequences of complex data in two real arrays containing the real and imaginary parts of the sequence respectively. The complex storage format routines store the sequences in complex arrays.
C06PUF and C06FXF/C06PXF should be used in preference to C06FJF/C06PJF for two- and three-dimensional transforms, as they are easier to use and are likely to be more efficient.

3.4.2  Real data

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 ${\stackrel{^}{z}}_{\left({n}_{1}-{k}_{1}\right){k}_{2}}$ are the complex conjugate of ${\stackrel{^}{z}}_{{k}_{1}{k}_{2}}$ for ${k}_{1}=0,1,\dots ,{n}_{1}/2$, and ${k}_{2}=0,1,\dots ,{n}_{2}-1$.

3.5  Convolution and Correlation

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.

3.6  Inverse Laplace Transforms

Two methods are provided: Weeks' method (C06LBF) and Crump's method (C06LAF). Both require the function $F\left(s\right)$ 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 $f\left(t\right)$ 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 $\alpha$ is close to $0$. For this purpose it is useful to remember that the inverse of $F\left(s+k\right)$ is $\mathrm{exp}\left(-kt\right)f\left(t\right)$. The method used by C06LAF is not so satisfactory when $f\left(t\right)$ is close to zero, in which case a term may be added to $F\left(s\right)$, e.g., $k/s+F\left(s\right)$ has the inverse $k+f\left(t\right)$.
Singularities in the inverse function $f\left(t\right)$ generally cause numerical methods to perform less well. The positions of singularities can often be identified by examination of $F\left(s\right)$. If $F\left(s\right)$ contains a term of the form $\mathrm{exp}\left(-ks\right)/s$ 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 $F\left(s\right)$ should be removed before computing the inverse.

3.7  Direct Summation of Orthogonal Series

The only routine available is C06DCF, which sums a finite Chebyshev series
 $∑ j=0 n cj Tj x , ∑ j=0 n cj T 2j x or ∑ j=0 n cj T 2j+1 x$
depending on the choice of a parameter.

3.8  Acceleration of Convergence

The only routine available is C06BAF.

4  Decision Trees

Tree 1: Fourier Transform of Discrete Complex Data

 Is the data one-dimensional? _yes Multiple vectors? _yes Stored as rows? _yes C06PRF | | no| | | Stored as columns? _yes C06PSF | no| | C06PCF no| Is the data two-dimensional? _yes C06PUF no| Is the data three-dimensional? _yes C06PXF no| Transform on one dimension only? _yes C06PFF no| Transform on all dimensions? _yes C06PJF

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

 Quarter-wave sine (inverse) transform _yes C06RCF no| Quarter-wave cosine (inverse) transform _yes C06RDF no| Sine (inverse) transform _yes C06RAF no| Cosine (inverse) transform _yes C06RBF no| Is the data three-dimensional? _yes Forward transform on real data _yes C06PYF | no| | Inverse transform on Hermitian data _yes C06PZF no| Is the data two-dimensional? _yes Forward transform on real data _yes C06PVF | no| | Inverse transform on Hermitian data _yes C06PWF no| Is the data multi one-dimensional? _yes Sequences stored by row _yes C06PPF | no| | Sequences stored by column _yes C06PQF no| Is the data one-dimensional? _yes C06PAF no| Transform in one-dimension only _yes C06FFF no| C06FJF

5  Functionality Index

 Acceleration of convergence C06BAF
 Convolution or Correlation,
 complex vectors C06PKF
 real vectors,
 time-saving C06FKF
 Discrete Fourier Transform,
 multidimensional,
 complex sequence,
 complex storage C06PJF
 real storage C06FJF
 multiple half- and quarter-wave transforms,
 Fourier cosine transforms,
 simple use C06RBF
 Fourier sine transforms,
 simple use C06RAF
 quarter-wave cosine transforms,
 simple use C06RDF
 quarter-wave sine transforms,
 simple use C06RCF
 one-dimensional,
 multiple transforms,
 complex sequence,
 complex storage by columns C06PSF
 complex storage by rows C06PRF
 Hermitian/real sequence,
 complex storage by columns C06PQF
 complex storage by rows C06PPF
 Hermitian sequence,
 real storage by rows C06FQF
 real sequence,
 real storage by rows C06FPF
 multi-variable,
 complex sequence,
 complex storage C06PFF
 real storage C06FFF
 single transforms,
 complex sequence,
 time-saving,
 complex storage C06PCF
 real storage C06FCF
 Hermitian/real sequence,
 time-saving,
 complex storage C06PAF
 Hermitian sequence,
 time-saving,
 real storage C06FBF
 real sequence,
 time-saving,
 real storage C06FAF
 three-dimensional,
 complex sequence,
 complex storage C06PXF
 real storage C06FXF
 Hermitian/real sequence,
 complex-to-real C06PZF
 real-to-complex C06PYF
 two-dimensional,
 complex sequence,
 complex storage C06PUF
 Hermitian/real sequence,
 complex-to-real C06PWF
 real-to-complex C06PVF
 Inverse Laplace Transform,
 Crump's method C06LAF
 Weeks' method,
 compute coefficients of solution C06LBF
 evaluate solution C06LCF
 Summation of Chebyshev series C06DCF

None.

7  Routines Withdrawn or Scheduled for Withdrawal

The following lists all those routines that have been withdrawn since Mark 17 of the Library or are scheduled for withdrawal at one of the next two marks.
 WithdrawnRoutine Mark ofWithdrawal Replacement Routine(s) C06DBF 25 C06DCF C06EAF 26 C06PAF C06EBF 26 C06PAF C06ECF 26 C06PCF C06EKF 26 C06FKF C06FRF 26 C06PSF C06FUF 26 C06PUF C06GBF 26 No replacement required C06GCF 26 No replacement required C06GQF 26 No replacement required C06GSF 26 No replacement required C06HAF 26 C06RAF C06HBF 26 C06RAF C06HCF 26 C06RCF C06HDF 26 C06RDF
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
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