This section describes some global information applying to the Fourier transform, convolution, and correlation subroutines.
This section provides some key points about using the Fourier transform, convolution, and correlation subroutines.
These subroutines use the term "sequences," rather than vectors and matrices, to describe the data that is stored in the arrays. The conventions used for representing sequences are defined in Sequences.
Some of the sequences used in these computations use a zero origin rather than a one-origin. For example, xj can be expressed with j = 0, 1, ..., n-1 rather than j = 1, 2, ..., n. When using the formulas provided in this book to calculate array sizes or offsets into arrays, you need to be careful that you substitute the correct values. For example, the number of xj elements in the sequence is n, not n-1.
The length of the transform you can use in your program depends on the limits of the addressability of your processor.
To determine acceptable lengths of the transforms in the Fourier transform subroutines, you have several choices. First, you can use the formula or table of values in Acceptable Lengths for the Transforms to choose a value. Second, ESSL's input-argument error recovery provides a means of determining an acceptable length of the transform. It uses the optionally-recoverable error 2030. For details, see Providing a Correct Transform Length to ESSL.
Use the following formula to determine acceptable transform lengths:
where:
Figure 13 lists all the acceptable values for transform lengths in the Fourier transform subroutines.
Figure 13. Table of Acceptable Lengths for the Transforms
2 4 6 8 10 12 14 16 18
20 22 24 28 30 32 36 40 42
44 48 56 60 64 66 70 72 80
84 88 90 96 110 112 120 126 128
132 140 144 154 160 168 176 180 192
198 210 220 224 240 252 256 264 280
288 308 320 330 336 352 360 384 396
420 440 448 462 480 504 512 528 560
576 616 630 640 660 672 704 720 768
770 792 840 880 896 924 960 990 1008
1024 1056 1120 1152 1232 1260 1280 1320 1344
1386 1408 1440 1536 1540 1584 1680 1760 1792
1848 1920 1980 2016 2048 2112 2240 2304 2310
2464 2520 2560 2640 2688 2772 2816 2880 3072
3080 3168 3360 3520 3584 3696 3840 3960 4032
4096 4224 4480 4608 4620 4928 5040 5120 5280
5376 5544 5632 5760 6144 6160 6336 6720 6930
7040 7168 7392 7680 7920 8064 8192 8448 8960
9216 9240 9856 10080 10240 10560 10752 11088 11264
11520 12288 12320 12672 13440 13860 14080 14336 14784
15360 15840 16128 16384 16896 17920 18432 18480 19712
20160 20480 21120 21504 22176 22528 23040 24576 24640
25344 26880 27720 28160 28672 29568 30720 31680 32256
32768 33792 35840 36864 36960 39424 40320 40960 42240
43008 44352 45056 46080 49152 49280 50688 53760 55440
56320 57344 59136 61440 63360 64512 65536 67584 71680
73728 73920 78848 80640 81920 84480 86016 88704 90112
92160 98304 98560 101376 107520 110880 112640 114688 118272
122880 126720 129024 131072 135168 143360 147456 147840 157696
161280 163840 168960 172032 177408 180224 184320 196608 197120
202752 215040 221760 225280 229376 236544 245760 253440 258048
262144 270336 286720 294912 295680 315392 322560 327680 337920
344064 354816 360448 368640 393216 394240 405504 430080 443520
450560 458752 473088 491520 506880 516096 524288 540672 573440
589824 591360 630784 645120 655360 675840 688128 709632 720896
737280 786432 788480 811008 860160 887040 901120 917504 946176
983040 1013760 1032192 1048576 1081344 1146880 1179648 1182720 1261568
1290240 1310720 1351680 1376256 1419264 1441792 1474560 1572864 1576960
1622016 1720320 1774080 1802240 1835008 1892352 1966080 2027520 2064384
2097152 2162688 2293760 2359296 2365440 2523136 2580480 2621440 2703360
2752512 2838528 2883584 2949120 3145728 3153920 3244032 3440640 3548160
3604480 3670016 3784704 3932160 4055040 4128768 4194304 4325376 4587520
4718592 4730880 5046272 5160960 5242880 5406720 5505024 5677056 5767168
5898240 6291456 6307840 6488064 6881280 7096320 7208960 7340032 7569408
7864320 8110080 8257536 8388608 8650752 9175040 9437184 9461760 10092544
10321920 10485760 10813440 11010048 11354112 11534336 11796480 12582912 12615680
12976128 13762560 14192640 14417920 14680064 15138816 15728640 16220160 16515072
16777216 17301504 18350080 18874368 18923520 20185088 20643840 20971520 21626880
22020096 22708224 23068672 23592960 25165824 25231360 25952256 27525120 28385280
28835840 29360128 30277632 31457280 32440320 33030144 33554432 34603008 36700160
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Auxiliary working storage is required by the Fourier transform subroutines and by the SCONF, SCORF, and SACORF subroutines. This storage is provided through the calling sequence arguments aux, aux1, and aux2. The sizes of these storage areas are specified by the calling sequence arguments naux, naux1, and naux2, respectively.
The aux1 array is used for storing tables and other parameters when you call a Fourier transform, convolution, or correlation subroutine for initialization with init = 1. The initialized aux1 array is then used on succeeding calls with init = 0, when the computation is actually done. You should not use this array between the initialization and the computation.
The aux and aux2 arrays are used for temporary storage during the running of the subroutine and are available for use by your program between calls to the subroutine.
The aux3 argument is provided for migration purposes only and is ignored.
In many of those subroutines requiring aux1 auxiliary working storage, two invocations of the subroutines are necessary. The first invocation initializes the working storage in aux1 for the subroutine, and the second performs the computations. (For an explanation of auxiliary working storage, see Understanding Auxiliary Working Storage Requirements.) As a result, the working storage in aux1 should not be used by the calling program between the two calls to the subroutine. However, it can be reused after intervening calls to the subroutine with different arguments.
If you plan to repeat a computation many times using the same set of arguments, you only need to do one initialization of the aux1 array; that is, the initialized aux1 array can be saved and reused as many times as needed for the computation.
If you plan to perform different computations, with different sets of arguments (except for input argument x), you need to do an initialization for each different computation; that is, you initialize the various aux1 arrays for use with the different computations, saving and reusing them until they are not needed any more.
To determine the size of auxiliary storage, you have several choices. First, you can use the formulas provided in each subroutine description. Second, ESSL's input-argument error recovery provides a means of determining the minimum size you need for auxiliary storage. It uses the optionally-recoverable error 2015. For details, see Using Auxiliary Storage in ESSL. Third, you can have ESSL dynamically allocate aux and aux2. For details, see Dynamic Allocation of Auxiliary Storage.