void	nag_sum_fast_gauss (Integer d, const double srcs[], Integer n, const double trgs[], Integer m, const double q[], Integer p1, Integer p2, Integer k, const double hin[], Integer lhin, double tol, double v[], double term[], NagError fail)

3

Description

nag_sum_fast_gauss (c06sac) calculates the

d

-dimensional fast Gauss transform (FGT),

\hat{G} (y)

, that approximates 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}

. The DGT 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).

This function implements the improved FGT algorithm presented in Raykar and Duraiswami (2005). The algorithm clusters the sources into

k

distinct clusters and then computes two Taylor series approximations per cluster with

p_{1}

and

p_{2}

terms respectively. You must provide

p_{1}

p_{2}

and

k

when calling the function. See Section 7 below for a further discussion on accuracy when choosing their values.

The input array

hin

of this function is designed to allow maximum flexibility in the supply of the standard deviation arguments by reusing, in a cyclic manner, elements of the array when it is less than

n

elements long. For example, if all Gaussian sources have the same standard deviation then it is only necessary to set

lhin

1

and to provide the value of the standard deviation in

hin (1)

; the function will then automatically expand

hin

to be of length

n

. For further details please see Section 2.6 in the g01 Chapter Introduction.

4

References

Greengard L and Strain J (1991) The Fast Gauss Transform SIAM J. Sci. Statist. Comput. 12(1) 79–94

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

5

Arguments

1: $d$ – IntegerInput: On entry: $d$ , the number of dimensions.

Constraint: $d > 0$ .
2: $srcs [d \times n]$ – const doubleInput: Note: the $(i, j)$ th element of the matrix is stored in $srcs [(j - 1) \times d + i - 1]$ .

On entry: $x$ , the locations of the Gaussian sources.
3: $n$ – IntegerInput: On entry: $n$ , the number of Gaussian sources.

Constraint: $n > 0$ .
4: $trgs [d \times m]$ – const doubleInput: Note: the $(i, j)$ th element of the matrix is stored in $trgs [(j - 1) \times d + i - 1]$ .

On entry: $y$ , the locations of the target points at which the FGT will be evaluated.
5: $m$ – IntegerInput: On entry: $m$ , the number of target points.

Constraint: $m > 0$ .
6: $q [n]$ – const doubleInput: On entry: $q$ , the weights of the Gaussian sources.
7: $p1$ – Integer *Input/Output: On entry: $p_{1}$ , the number of terms of the first Taylor series to be evaluated.

On exit: p1 is unchanged.

Constraint: $p1 > 0$ .
8: $p2$ – Integer *Input/Output: On entry: $p_{2}$ , the number of terms of the second Taylor series to be evaluated.

On exit: p2 is unchanged.

Constraint: $p2 > 0$ .
9: $k$ – Integer *Input/Output: On entry: $k$ , the number of clusters into which the source points will be aggregated.

On exit: k is unchanged.

Constraint: $1 \leq k \leq n$ .
10: $hin [lhin]$ – const doubleInput: On entry: $h$ , the standard deviations of the Gaussian sources. If $lhin < n$ , the array will be expanded automatically by repeating hin until it is of length n. See Section 2.6 in the g01 Chapter Introduction for further information.

Constraint: $hin [i] > 0.0$ , for $i = 0, 1, \dots, lhin - 1$ .
11: $lhin$ – IntegerInput: On entry: the length of the array hin.

Constraint: $1 \leq lhin \leq n$ .
12: $tol$ – doubleInput: On entry: $ε$ , the desired accuracy of the FGT approximation of the DGT. Determines the radius of the source clusters: the contribution of a source point to the FGT approximation at a target point is disregarded if the source is outside the corresponding cluster radius.

Constraint: $tol > 0.0$ .
13: $v [m]$ – doubleOutput: On exit: $\hat{G} (y)$ , the value of the FGT evaluated at $y$ .
14: $term [m]$ – doubleOutput: On exit: $term [j - 1]$ contains the absolute value of the final Taylor series term that is largest, relative to the size of the sum of the corresponding series, across all clusters that contribute to the FGT at target point $v [j - 1]$ .
15: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error 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_SIZE: On entry, $lhin = 〈value〉$ .
Constraint: $1 \leq lhin \leq n$ .
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $d = 〈value〉$ .
Constraint: $d > 0$ .

On entry, $m = 〈value〉$ .
Constraint: $m > 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .

On entry, $p1 = 〈value〉$ .
Constraint: $p1 > 0$ .

On entry, $p2 = 〈value〉$ .
Constraint: $p2 > 0$ .
NE_INT_2: On entry, $k = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 \leq k \leq n$ .
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.
NE_REAL: On entry, $tol = 〈value〉$ .
Constraint: $tol > 0.0$ .
NE_REAL_ARRAY_INPUT: On entry, $hin [〈value〉] = 〈value〉$ .
Constraint: $hin [i] > 0.0$ , for $i = 0, 1, \dots, lhin - 1$ .
NW_TOO_FEW_TERMS: On exit, $p1 = 〈value〉$ , $p2 = 〈value〉$ and $k = 〈value〉$ .
p1, p2 or k may have been too small to calculate $v [m - 1]$ to the required accuracy tol.

7

Accuracy

The function does not currently implement the procedure described in Raykar and Duraiswami (2005) for automatically determining values for p1, p2 and k. Non-zero values must therefore be provided for these parameters when calling the function.

For a given set of source and target points and a specified tolerance, there is an interaction between the number of clusters, k, and the number of Taylor series terms, p1 and p2: if the sources are clustered together in fewer clusters (small k) then more terms will be needed in each cluster's Taylor series (large p1 and p2) to capture the effect of the source points further from the cluster centres. Increasing the number of clusters reduces their individual radii and requires fewer terms in their Taylor series, but increases the number of Taylor series that must be evaluated overall.

If the source and target points are uniformly distributed in a unit hypercube, Raykar and Duraiswami (2005) advise users to select

k \sim ⌈{(h_{\max} + h_{\min} / 2)}^{- d}⌉

. If the points are not uniformly distributed then more clusters than this will be needed to calculate the FGT to within the specified tol without requiring prohibitively large values for p1 and p2.

There is less guidance available for selecting good values for p1 and p2. As the number of Taylor series terms is a major factor on the computation time taken by this function, you are advised to initially try a small number, e.g.

20

or so, and then tune p1 and p2 up or down based on the values returned. Note that p1 and p2 are not required to have identical values.

To aid the selection of values for p1, p2 and k, the function returns in

term [j - 1]

the absolute value of the final Taylor series term that is largest, relative to the size of the sum of the corresponding series, across all clusters that contribute to the FGT at target point

j

. If this value is larger than the requested tol, the function will additionally return a non-zero fail value and you are advised to re-run the function with larger p1, p2 or k.

8

Parallelism and Performance

nag_sum_fast_gauss (c06sac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_sum_fast_gauss (c06sac) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The time complexity of the algorithm implemented by this function is

O (M + N)

, versus the

O (M N)

time complexity of evaluating the DGT directly.

10

Example

In this example values for

x

y

p_{1}

p_{2}

k

and

ε

are read in,

\hat{G} (y)

calculated and the results displayed.

NAG Library Function Document

nag_sum_fast_gauss (c06sac)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results