NAG Library Routine Document

G01AMF

G01AMF uses a modified version of Singleton's ‘median-of-three’ Quicksort algorithm (Singleton (1969)) to determine specified quantiles of a vector of real values. The input vector is partially sorted, as far as is required to compute desired quantiles; for a single quantile, this is much faster than sorting the entire vector. Where necessary, linear interpolation is also carried out to return the values of quantiles which lie between two data points.

4 References

Singleton R C (1969) An efficient algorithm for sorting with minimal storage: Algorithm 347 Comm. ACM 12 185–187

5 Parameters

1: N – INTEGERInput

On entry: the number of elements in the input vector RV.

Constraint: N>0.

2: RV(N) – double precision arrayInput/Output

On entry: the vector whose quantiles are to be determined.

On exit: the order of the elements in RV is not, in general, preserved.

3: NQ – INTEGERInput

On entry: the number of quantiles requested.

Constraint: NQ>0.

4: Q(NQ) – double precision arrayInput

On entry: the quantiles to be calculated, in ascending order. Note that these must be between 0.0 and 1.0, with 0.0 returning the smallest element and 1.0 the largest.

Constraints:

0.0≤Qi≤1.0, for i=1,2,…,NQ;
Qi≤Qi+1, for i=1,2,…,NQ-1.

5: QV(NQ) – double precision arrayOutput

On exit: QVi contains the quantile specified by the value provided in Qi, or an interpolated value if the quantile falls between two data values.

6: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to 0, -1 or 1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

On exit: IFAIL=0 unless the routine detects an error (see Section 6).

For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.