# NAG C Library Function Document

## 1Purpose

nag_imodwt (c09dbc) computes the inverse one-dimensional maximal overlap discrete wavelet transform (MODWT) at a single level. The initialization function nag_wfilt (c09aac) must be called first to set up the MODWT options.

## 2Specification

 #include #include
 void nag_imodwt (Integer lenc, const double ca[], const double cd[], Integer n, double y[], const Integer icomm[], NagError *fail)

## 3Description

nag_imodwt (c09dbc) performs the inverse operation of nag_modwt (c09dac). That is, given sets of ${n}_{c}$ approximation coefficients and detail coefficients, computed by nag_modwt (c09dac) using a MODWT as set up by the initialization function nag_wfilt (c09aac), on a real data array of length $n$, nag_imodwt (c09dbc) will reconstruct the data array ${y}_{i}$, for $\mathit{i}=1,2,\dots ,n$, from which the coefficients were derived.

## 4References

Percival D B and Walden A T (2000) Wavelet Methods for Time Series Analysis Cambridge University Press

## 5Arguments

1:    $\mathbf{lenc}$IntegerInput
On entry: the dimension of the arrays ca and cd.
Constraint: ${\mathbf{lenc}}\ge {n}_{c}$, where ${n}_{c}$ is the value returned in nwc by the call to the initialization function nag_wfilt (c09aac).
2:    $\mathbf{ca}\left[{\mathbf{lenc}}\right]$const doubleInput
On entry: the ${n}_{c}$ approximation coefficients, ${C}_{a}$. These will normally be the result of some transformation on the coefficients computed by nag_modwt (c09dac).
3:    $\mathbf{cd}\left[{\mathbf{lenc}}\right]$const doubleInput
On entry: the ${n}_{c}$ detail coefficients, ${C}_{d}$. These will normally be the result of some transformation on the coefficients computed by nag_modwt (c09dac).
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the length of the original data array from which the wavelet coefficients were computed by nag_modwt (c09dac) and the length of the data array y that is to be reconstructed by this function.
Constraint: This must be the same as the value n passed to the initialization function nag_wfilt (c09aac).
5:    $\mathbf{y}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the reconstructed data based on approximation and detail coefficients ${C}_{a}$ and ${C}_{d}$ and the transform options supplied to the initialization function nag_wfilt (c09aac).
6:    $\mathbf{icomm}\left[100\right]$const IntegerCommunication Array
On entry: contains details of the discrete wavelet transform and the problem dimension and, possibly, additional information on the previously computed forward transform.
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ARRAY_DIM_LEN
On entry, array dimension lenc not large enough: ${\mathbf{lenc}}=〈\mathit{\text{value}}〉$ but must be at least $〈\mathit{\text{value}}〉$.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INITIALIZATION
On entry, n is inconsistent with the value passed to the initialization function: ${\mathbf{n}}=〈\mathit{\text{value}}〉$, n should be $〈\mathit{\text{value}}〉$.
On entry, the initialization function nag_wfilt (c09aac) has not been called first or it has not been called with ${\mathbf{wtrans}}=\mathrm{Nag_MODWTSingle}$, or the communication array icomm has become corrupted.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The accuracy of the wavelet transform depends only on the floating-point operations used in the convolution and downsampling and should thus be close to machine precision.

## 8Parallelism and Performance

nag_imodwt (c09dbc) is not threaded in any implementation.