1378:
positions in a one-dimensional array. Of course, deviating from convention probably incurs a cost that increases with the degree of necessary interaction with conventional language features and other code, not only in the form of increased vulnerability to mistakes (forgetting to also invert matrix multiplication order, reverting to convention during code maintenance, etc.), but also in the form of having to actively rearrange elements, all of which have to be weighed against any original purpose such as increasing performance. Running the loop row-wise is preferred in row-major languages like C and vice versa for column-major languages.
31:
1564:, which was to write vectors as rows; for compatibility, transformation matrices would still be stored in vector-major (=row-major) rather than coordinate-major (=column-major) order, and he then used the trick " say that matrices in OpenGL are stored in column-major order". This was really only relevant for presentation, because matrix multiplication was stack-based and could still be interpreted as post-multiplication, but, worse, reality leaked through the C-based
353:
2087:
2322:
194:
index, no grouping order between the dimensions is implied by this. The choice of how to group and order the indices, either by row-major or column-major methods, is thus a matter of convention. The same terminology can be applied to even higher dimensional arrays. Row-major grouping starts from the
2631:
An array-type that specifies a sequence of two or more index-types shall be an abbreviated notation for an array-type specified to have as its index-type the first index-type in the sequence and to have a component-type that is an array-type specifying the sequence of index-types without the first
57:
The difference between the orders lies in which elements of an array are contiguous in memory. In row-major order, the consecutive elements of a row reside next to each other, whereas the same holds true for consecutive elements of a column in column-major order. While the terms allude to the rows
1606:
is typically an expensive operation, some systems provide options to specify individual matrices as being stored transposed. The programmer must then decide whether or not to rearrange the elements in memory, based on the actual usage (including the number of times that the array is reused in a
102:
The terms row-major and column-major stem from the terminology related to ordering objects. A general way to order objects with many attributes is to first group and order them by one attribute, and then, within each such group, group and order them by another attribute, etc. If more than one
1377:
To use column-major order in a row-major environment, or vice versa, for whatever reason, one workaround is to assign non-conventional roles to the indexes (using the first index for the column and the second index for the row), and another is to bypass language syntax by explicitly computing
73:
Data layout is critical for correctly passing arrays between programs written in different programming languages. It is also important for performance when traversing an array because modern CPUs process sequential data more efficiently than nonsequential data. This is primarily due to
216:
66:, respectively. It is also worth noting that matrices, being commonly represented as collections of row or column vectors, using this approach are effectively stored as consecutive vectors or consecutive vector components. Such ways of storing data are referred to as
1845:
2457:
of dimensions (with row-major and column-order just 2 special cases), although the lists of stride values are not necessarily permutations of each other, e.g., in the 2-by-3 example above, the strides are (3,1) for row-major and (1,2) for column-major.
2098:
2660:
From right to left, the rightmost dimension represents columns; the next dimension represents rows. SAS places variables into a multidimensional array by filling all rows in order, beginning at the upper left corner of the array (known as row-major
1530:
Support for multi-dimensional arrays may also be provided by external libraries, which may even support arbitrary orderings, where each dimension has a stride value, and row-major or column-major are just two possible resulting interpretations.
1584:). As a result, many developers will now simply declare that having the column as the first index is the definition of column-major, even though this is clearly not the case with a real column-major language like Fortran.
1560:) for graphics processing. Since "recent mathematical treatments of linear algebra and related fields invariably treat vectors as columns," designer Mark Segal decided to substitute this for the convention in predecessor
2424:
1686:
348:{\displaystyle A=a_{y,x}={\begin{bmatrix}\color {Blue}a_{11}&\color {Blue}a_{12}&\color {Blue}a_{13}\\\color {Orange}a_{21}&\color {Orange}a_{22}&\color {Orange}a_{23}\end{bmatrix}}}
2676:
1370:
row part can even be assigned to an intermediate variable that is then indexed in a separate expression. (No other implications should be assumed, e.g., Fortran is not column-major simply
1771:
759:
729:
664:
629:
599:
499:
694:
564:
534:
469:
434:
404:
2082:{\displaystyle n_{d}+N_{d}\cdot (n_{d-1}+N_{d-1}\cdot (n_{d-2}+N_{d-2}\cdot (\cdots +N_{2}n_{1})\cdots ))=\sum _{k=1}^{d}\left(\prod _{\ell =k+1}^{d}N_{\ell }\right)n_{k}}
1386:
Programming languages or their standard libraries that support multi-dimensional arrays typically have a native row-major or column-major storage order for these arrays.
1833:
1119:
825:
177:
2317:{\displaystyle n_{1}+N_{1}\cdot (n_{2}+N_{2}\cdot (n_{3}+N_{3}\cdot (\cdots +N_{d-1}n_{d})\cdots ))=\sum _{k=1}^{d}\left(\prod _{\ell =1}^{k-1}N_{\ell }\right)n_{k}}
1347:
1309:
1271:
1233:
1195:
1157:
1053:
1015:
977:
939:
901:
863:
1476:, which typically store pointers to elements in the same row contiguously (like row-major order), but not the rows themselves. They are used in (ordered by age):
144:
2611:
Initial values specified for an array are assigned to successive elements of the array in row-major order (final subscript varying most rapidly).
2335:
770:, multidimensional arrays are stored in row-major order, and the array indexes are written row-first (lexicographical access order):
2472:
2812:
Because Scilab stores arrays in column major format, the elements of a column are adjacent (i.e. a separation of 1) in linear format.
2754:
179:
is in its first row and second column. This convention is carried over to the syntax in programming languages, although often with
2595:
1632:
1064:, arrays are stored in column-major order, while the array indexes are still written row-first (colexicographical access order):
62:, the orders can be generalized to arrays of any dimension by noting that the terms row-major and column-major are equivalent to
1602:, an array stored as row-major but read as column-major (or vice versa) will appear transposed. As actually performing this
1410:
1362:
as in
Fortran, almost inevitably implies row-major order for syntactic reasons, so to speak, because it can be rewritten as
83:
1611:
1576:, which unfortunately muddled the convention that the designer sought to adopt, and this was even preserved in the
3107:
1711:
17:
2704:
1485:
1481:
1603:
1504:
1457:
1406:
3057:
1497:
1493:
1437:
735:
705:
640:
605:
575:
475:
2762:(Version 1.4 ed.), Cambridge, UK: MRC Biostatistics Unit, Institute of Public Health, archived from
2570:
2500:
2491:, another way of mapping multidimensional data to a one-dimensional index, useful in tree data structures
1477:
670:
540:
510:
445:
410:
380:
2644:
1580:
that was later added (although this also makes it possible to access coordinates by name instead, e.g.,
1374:
of its notation, and even the above implication could intentionally be circumvented in a new language.)
1519:
1425:
87:
2494:
115:
1587:
1449:
1441:
1390:
767:
2917:
3015:
2988:
1577:
3003:
If the storage order is not specified, then Eigen defaults to storing the entry in column-major.
2849:
2632:
index-type in the sequence and specifying the same component-type as the original specification.
2545:
111:. If two attributes participate in ordering, it is sufficient to name only the major attribute.
2467:
1780:
47:
2963:
2783:
2483:
1542:
1091:
797:
149:
79:
1325:
1287:
1249:
1211:
1173:
1135:
1031:
993:
955:
917:
879:
841:
3036:
2824:
2521:
180:
59:
51:
8:
2763:
2746:
1512:
204:
63:
2871:
2796:
2430:
129:
91:
2896:
2329:
1508:
2717:
3125:
2623:
1068:
Fortran: column-major order (colexicographical access order), one-based indexing
2095:
dimension is contiguous, so that the memory-offset of this element is given by:
1842:
dimension is contiguous, so that the memory-offset of this element is given by:
114:
In the case of arrays, the attributes are the indices along each dimension. For
2503:, the equivalent of turning a matrix into the corresponding column-major vector
1489:
3119:
2325:
1473:
30:
2419:{\textstyle \prod _{\ell =1}^{0}N_{\ell }=\prod _{\ell =d+1}^{d}N_{\ell }=1}
2488:
2477:
1414:
2454:
1546:
1398:
67:
1358:
with multi-step indexing as in C, as opposed to a neutral notation like
2750:
1433:
774:
C: row-major order (lexicographical access order), zero-based indexing
2480:, another difference between array types across programming languages
1599:
1557:
75:
2942:
86:
instructions that operate on vectors of data. In some media such as
1614:
functions are passed flags indicating which arrays are transposed.
1461:
2753:; Lunn, Dave (January 2003), "Formatting of data: S-Plus format",
2437:
is given by the multiplication value in parentheses before index
1622:
The concept generalizes to arrays with more than two dimensions.
1561:
1421:
1061:
34:
Illustration of difference between row- and column-major ordering
3112:, third edition, section 2.2.6 (Addison-Wesley: New York, 1997).
1590:(for Lua) changed from column-major to row-major default order.
3082:
1553:
1453:
1445:
1429:
103:
attribute participates in ordering, the first would be called
1706:
1535:
1394:
2745:
2741:
1681:{\displaystyle N_{1}\times N_{2}\times \cdots \times N_{d}}
1402:
82:. In addition, contiguous access makes it possible to use
1565:
1598:
As exchanging the indices of an array is the essence of
1518:
An alternative approach uses tables of tables, e.g., in
1472:
A typical alternative for dense array storage is to use
766:
Programming languages handle this in different ways. In
118:
in mathematical notation, the first index indicates the
2645:"SAS® 9.4 Language Reference: Concepts, Sixth Edition"
2338:
1381:
250:
2101:
1848:
1783:
1714:
1635:
1467:
1328:
1290:
1252:
1214:
1176:
1138:
1094:
1034:
996:
958:
920:
882:
844:
800:
738:
708:
673:
643:
608:
578:
543:
513:
478:
448:
413:
383:
219:
152:
132:
2650:. SAS Institute Inc. September 6, 2017. p. 573
2497:, a technique for storing sparse matrices in memory
1705:), a given element of this array is specified by a
1503:Even less dense is to use lists of lists, e.g., in
205:
lexicographic and colexicographic (or colex) orders
2418:
2316:
2081:
1827:
1765:
1680:
1341:
1303:
1265:
1227:
1189:
1151:
1113:
1047:
1009:
971:
933:
895:
857:
819:
753:
723:
688:
658:
623:
593:
558:
528:
493:
463:
428:
398:
347:
171:
138:
1617:
1568:because individual elements would be accessed as
3117:
2514:
2453:possible orders for a given array, one for each
2707:(retrieved from Mathworks.co.uk, January 2014).
58:and columns of a two-dimensional array, i.e. a
2943:"11.2 – Matrices and Multi-Dimensional Arrays"
2897:"The Python Standard Library: 8. Data Types"
2825:"Internal array representation in rasdaman"
2473:Comparison of programming languages (array)
1766:{\displaystyle (n_{1},n_{2},\ldots ,n_{d})}
97:
3083:"BLAS (Basic Linear Algebra Subprograms)"
2446:in the right-hand side summations above.
2699:
2697:
2596:"Language Reference Version 4 Release 3"
186:Even though the row is indicated by the
64:lexicographic and colexicographic orders
29:
2735:
27:Array representation in computer memory
14:
3118:
358:could be stored in two possible ways:
2694:
2671:
2669:
1541:Column-major order is the default in
1525:
754:{\displaystyle \color {Orange}a_{23}}
739:
724:{\displaystyle \color {Orange}a_{23}}
709:
674:
659:{\displaystyle \color {Orange}a_{22}}
644:
624:{\displaystyle \color {Orange}a_{22}}
609:
594:{\displaystyle \color {Orange}a_{21}}
579:
544:
514:
494:{\displaystyle \color {Orange}a_{21}}
479:
449:
414:
384:
325:
311:
297:
281:
267:
253:
2571:"Why numbering should start at zero"
2964:"The N-dimensional array (ndarray)"
2677:"Columns, Rows, and Array Majority"
1382:Programming languages and libraries
689:{\displaystyle \color {Blue}a_{13}}
559:{\displaystyle \color {Blue}a_{12}}
529:{\displaystyle \color {Blue}a_{13}}
464:{\displaystyle \color {Blue}a_{12}}
429:{\displaystyle \color {Blue}a_{11}}
399:{\displaystyle \color {Blue}a_{11}}
24:
2666:
1534:Row-major order is the default in
1468:Neither row-major nor column-major
94:faster than nonsequential access.
25:
3137:
2797:"FFTs with multidimensional data"
3110:Volume 1: Fundamental Algorithms
3016:"Column Vectors Vs. Row Vectors"
1612:Basic Linear Algebra Subprograms
1593:
199:index and column-major from the
3108:The Art of Computer Programming
3075:
3050:
3029:
3008:
2981:
2956:
2935:
2910:
2889:
2864:
2842:
2817:
2789:
2773:
122:, and the second indicates the
3062:Torch Package Reference Manual
2710:
2637:
2616:
2588:
2563:
2538:
2230:
2227:
2221:
2186:
2157:
2128:
1995:
1992:
1986:
1957:
1916:
1875:
1822:
1797:
1760:
1715:
1618:Address calculation in general
1420:Column-major order is used in
13:
1:
2850:"Java Language Specification"
2681:www.nv5geospatialsoftware.com
2507:
190:index and the column by the
90:, accessing sequentially is
7:
2501:Vectorization (mathematics)
2461:
2091:In column-major order, the
1389:Row-major order is used in
10:
3142:
3099:
2742:Spiegelhalter et al. (2003
2718:"Multi-dimensional Arrays"
2546:"Arrays and Formatted I/O"
2449:More generally, there are
88:magnetic-tape data storage
50:in linear storage such as
1828:{\displaystyle n_{k}\in }
2756:WinBUGS User Manual
1838:In row-major order, the
1552:A special case would be
46:are methods for storing
2989:"Eigen: Storage orders"
2786:(retrieved March 2010).
2429:For a given order, the
1604:rearrangement in memory
1578:OpenGL Shading Language
1114:{\displaystyle a_{y,x}}
820:{\displaystyle a_{y,x}}
210:For example, the array
172:{\displaystyle a_{1,2}}
126:, e.g., given a matrix
98:Explanation and example
78:which exploits spatial
48:multidimensional arrays
2918:"Vectors and Matrices"
2876:Scala Standard Library
2703:MATLAB documentation,
2624:"ISO/IEC 7185:1990(E)"
2575:E. W. Dijkstra Archive
2468:Array (data structure)
2420:
2399:
2359:
2328:is the multiplicative
2318:
2288:
2256:
2083:
2053:
2021:
1829:
1767:
1688:array with dimensions
1682:
1401:(for C-style arrays),
1343:
1342:{\displaystyle a_{23}}
1305:
1304:{\displaystyle a_{13}}
1267:
1266:{\displaystyle a_{22}}
1229:
1228:{\displaystyle a_{12}}
1191:
1190:{\displaystyle a_{21}}
1153:
1152:{\displaystyle a_{11}}
1115:
1060:On the other hand, in
1049:
1048:{\displaystyle a_{23}}
1011:
1010:{\displaystyle a_{22}}
973:
972:{\displaystyle a_{21}}
935:
934:{\displaystyle a_{13}}
897:
896:{\displaystyle a_{12}}
859:
858:{\displaystyle a_{11}}
821:
755:
725:
690:
660:
625:
595:
560:
530:
495:
465:
430:
400:
349:
173:
140:
35:
2484:Matrix representation
2421:
2373:
2339:
2319:
2262:
2236:
2084:
2027:
2001:
1830:
1777:(zero-based) indices
1768:
1683:
1344:
1306:
1268:
1230:
1192:
1154:
1116:
1050:
1012:
974:
936:
898:
860:
822:
756:
726:
691:
661:
626:
596:
561:
531:
496:
466:
431:
401:
350:
181:indexes starting at 0
174:
141:
80:locality of reference
33:
2780:An Introduction to R
2747:Spiegelhalter, David
2336:
2099:
1846:
1781:
1712:
1633:
1354:Note how the use of
1326:
1288:
1250:
1212:
1174:
1136:
1092:
1032:
994:
956:
918:
880:
842:
798:
736:
706:
671:
641:
606:
576:
541:
511:
476:
446:
411:
381:
217:
150:
130:
52:random access memory
2993:eigen.tuxfamily.org
2784:Section 5.1: Arrays
2705:MATLAB Data Storage
1600:array transposition
1513:Wolfram Mathematica
1069:
775:
370:Column-major order
92:orders of magnitude
2749:; Thomas, Andrew;
2416:
2314:
2079:
1825:
1763:
1678:
1526:External libraries
1339:
1301:
1263:
1225:
1187:
1149:
1111:
1067:
1045:
1007:
969:
931:
893:
855:
817:
773:
751:
750:
721:
720:
686:
685:
656:
655:
621:
620:
591:
590:
556:
555:
526:
525:
491:
490:
461:
460:
426:
425:
396:
395:
345:
339:
336:
322:
308:
292:
278:
264:
203:index, leading to
169:
136:
44:column-major order
36:
3105:Donald E. Knuth,
2526:Peter Lars Dordal
1610:For example, the
1572:or, effectively,
1352:
1351:
1058:
1057:
764:
763:
139:{\displaystyle A}
16:(Redirected from
3133:
3093:
3092:
3090:
3089:
3079:
3073:
3072:
3070:
3068:
3054:
3048:
3047:
3045:
3043:
3033:
3027:
3026:
3024:
3022:
3012:
3006:
3005:
3000:
2999:
2985:
2979:
2978:
2976:
2974:
2960:
2954:
2953:
2951:
2949:
2939:
2933:
2932:
2930:
2928:
2914:
2908:
2907:
2905:
2903:
2893:
2887:
2886:
2884:
2882:
2868:
2862:
2861:
2859:
2857:
2846:
2840:
2839:
2837:
2835:
2821:
2815:
2814:
2809:
2807:
2793:
2787:
2777:
2771:
2770:
2768:
2761:
2739:
2733:
2732:
2730:
2728:
2714:
2708:
2701:
2692:
2691:
2689:
2687:
2673:
2664:
2663:
2657:
2655:
2649:
2641:
2635:
2634:
2628:
2620:
2614:
2613:
2608:
2606:
2600:
2592:
2586:
2585:
2583:
2581:
2567:
2561:
2560:
2558:
2556:
2550:FORTRAN Tutorial
2542:
2536:
2535:
2533:
2532:
2518:
2425:
2423:
2422:
2417:
2409:
2408:
2398:
2393:
2369:
2368:
2358:
2353:
2330:identity element
2323:
2321:
2320:
2315:
2313:
2312:
2303:
2299:
2298:
2297:
2287:
2276:
2255:
2250:
2220:
2219:
2210:
2209:
2182:
2181:
2169:
2168:
2153:
2152:
2140:
2139:
2124:
2123:
2111:
2110:
2088:
2086:
2085:
2080:
2078:
2077:
2068:
2064:
2063:
2062:
2052:
2047:
2020:
2015:
1985:
1984:
1975:
1974:
1953:
1952:
1934:
1933:
1912:
1911:
1893:
1892:
1871:
1870:
1858:
1857:
1834:
1832:
1831:
1826:
1815:
1814:
1793:
1792:
1772:
1770:
1769:
1764:
1759:
1758:
1740:
1739:
1727:
1726:
1687:
1685:
1684:
1679:
1677:
1676:
1658:
1657:
1645:
1644:
1583:
1575:
1571:
1549:(both for C++).
1509:Wolfram Language
1369:
1365:
1361:
1357:
1348:
1346:
1345:
1340:
1338:
1337:
1320:
1310:
1308:
1307:
1302:
1300:
1299:
1282:
1272:
1270:
1269:
1264:
1262:
1261:
1244:
1234:
1232:
1231:
1226:
1224:
1223:
1206:
1196:
1194:
1193:
1188:
1186:
1185:
1168:
1158:
1156:
1155:
1150:
1148:
1147:
1130:
1120:
1118:
1117:
1112:
1110:
1109:
1084:
1077:
1070:
1066:
1054:
1052:
1051:
1046:
1044:
1043:
1026:
1016:
1014:
1013:
1008:
1006:
1005:
988:
978:
976:
975:
970:
968:
967:
950:
940:
938:
937:
932:
930:
929:
912:
902:
900:
899:
894:
892:
891:
874:
864:
862:
861:
856:
854:
853:
836:
826:
824:
823:
818:
816:
815:
790:
783:
776:
772:
760:
758:
757:
752:
749:
748:
730:
728:
727:
722:
719:
718:
695:
693:
692:
687:
684:
683:
665:
663:
662:
657:
654:
653:
630:
628:
627:
622:
619:
618:
600:
598:
597:
592:
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588:
565:
563:
562:
557:
554:
553:
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523:
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467:
462:
459:
458:
435:
433:
432:
427:
424:
423:
405:
403:
402:
397:
394:
393:
367:Row-major order
361:
360:
354:
352:
351:
346:
344:
343:
335:
334:
321:
320:
307:
306:
291:
290:
277:
276:
263:
262:
241:
240:
207:, respectively.
178:
176:
175:
170:
168:
167:
145:
143:
142:
137:
21:
3141:
3140:
3136:
3135:
3134:
3132:
3131:
3130:
3116:
3115:
3102:
3097:
3096:
3087:
3085:
3081:
3080:
3076:
3066:
3064:
3056:
3055:
3051:
3041:
3039:
3035:
3034:
3030:
3020:
3018:
3014:
3013:
3009:
2997:
2995:
2987:
2986:
2982:
2972:
2970:
2962:
2961:
2957:
2947:
2945:
2941:
2940:
2936:
2926:
2924:
2916:
2915:
2911:
2901:
2899:
2895:
2894:
2890:
2880:
2878:
2870:
2869:
2865:
2855:
2853:
2848:
2847:
2843:
2833:
2831:
2823:
2822:
2818:
2805:
2803:
2795:
2794:
2790:
2778:
2774:
2766:
2759:
2744:, p. 17):
2740:
2736:
2726:
2724:
2716:
2715:
2711:
2702:
2695:
2685:
2683:
2675:
2674:
2667:
2653:
2651:
2647:
2643:
2642:
2638:
2626:
2622:
2621:
2617:
2604:
2602:
2598:
2594:
2593:
2589:
2579:
2577:
2569:
2568:
2564:
2554:
2552:
2544:
2543:
2539:
2530:
2528:
2520:
2519:
2515:
2510:
2464:
2445:
2404:
2400:
2394:
2377:
2364:
2360:
2354:
2343:
2337:
2334:
2333:
2308:
2304:
2293:
2289:
2277:
2266:
2261:
2257:
2251:
2240:
2215:
2211:
2199:
2195:
2177:
2173:
2164:
2160:
2148:
2144:
2135:
2131:
2119:
2115:
2106:
2102:
2100:
2097:
2096:
2073:
2069:
2058:
2054:
2048:
2031:
2026:
2022:
2016:
2005:
1980:
1976:
1970:
1966:
1942:
1938:
1923:
1919:
1901:
1897:
1882:
1878:
1866:
1862:
1853:
1849:
1847:
1844:
1843:
1810:
1806:
1788:
1784:
1782:
1779:
1778:
1754:
1750:
1735:
1731:
1722:
1718:
1713:
1710:
1709:
1696:
1672:
1668:
1653:
1649:
1640:
1636:
1634:
1631:
1630:
1620:
1596:
1581:
1573:
1569:
1528:
1470:
1384:
1367:
1363:
1359:
1355:
1333:
1329:
1327:
1324:
1323:
1318:
1295:
1291:
1289:
1286:
1285:
1280:
1257:
1253:
1251:
1248:
1247:
1242:
1219:
1215:
1213:
1210:
1209:
1204:
1181:
1177:
1175:
1172:
1171:
1166:
1143:
1139:
1137:
1134:
1133:
1128:
1099:
1095:
1093:
1090:
1089:
1088:
1082:
1081:
1075:
1074:
1039:
1035:
1033:
1030:
1029:
1024:
1001:
997:
995:
992:
991:
986:
963:
959:
957:
954:
953:
948:
925:
921:
919:
916:
915:
910:
887:
883:
881:
878:
877:
872:
849:
845:
843:
840:
839:
834:
805:
801:
799:
796:
795:
794:
788:
787:
781:
780:
744:
740:
737:
734:
733:
714:
710:
707:
704:
703:
679:
675:
672:
669:
668:
649:
645:
642:
639:
638:
614:
610:
607:
604:
603:
584:
580:
577:
574:
573:
549:
545:
542:
539:
538:
519:
515:
512:
509:
508:
484:
480:
477:
474:
473:
454:
450:
447:
444:
443:
419:
415:
412:
409:
408:
389:
385:
382:
379:
378:
338:
337:
330:
326:
323:
316:
312:
309:
302:
298:
294:
293:
286:
282:
279:
272:
268:
265:
258:
254:
246:
245:
230:
226:
218:
215:
214:
157:
153:
151:
148:
147:
131:
128:
127:
100:
40:row-major order
28:
23:
22:
18:Row-major order
15:
12:
11:
5:
3139:
3129:
3128:
3114:
3113:
3101:
3098:
3095:
3094:
3074:
3049:
3028:
3007:
2980:
2955:
2934:
2909:
2888:
2872:"object Array"
2863:
2841:
2816:
2788:
2772:
2734:
2709:
2693:
2665:
2636:
2615:
2587:
2562:
2537:
2522:"Cache Memory"
2512:
2511:
2509:
2506:
2505:
2504:
2498:
2492:
2486:
2481:
2475:
2470:
2463:
2460:
2441:
2415:
2412:
2407:
2403:
2397:
2392:
2389:
2386:
2383:
2380:
2376:
2372:
2367:
2363:
2357:
2352:
2349:
2346:
2342:
2311:
2307:
2302:
2296:
2292:
2286:
2283:
2280:
2275:
2272:
2269:
2265:
2260:
2254:
2249:
2246:
2243:
2239:
2235:
2232:
2229:
2226:
2223:
2218:
2214:
2208:
2205:
2202:
2198:
2194:
2191:
2188:
2185:
2180:
2176:
2172:
2167:
2163:
2159:
2156:
2151:
2147:
2143:
2138:
2134:
2130:
2127:
2122:
2118:
2114:
2109:
2105:
2076:
2072:
2067:
2061:
2057:
2051:
2046:
2043:
2040:
2037:
2034:
2030:
2025:
2019:
2014:
2011:
2008:
2004:
2000:
1997:
1994:
1991:
1988:
1983:
1979:
1973:
1969:
1965:
1962:
1959:
1956:
1951:
1948:
1945:
1941:
1937:
1932:
1929:
1926:
1922:
1918:
1915:
1910:
1907:
1904:
1900:
1896:
1891:
1888:
1885:
1881:
1877:
1874:
1869:
1865:
1861:
1856:
1852:
1824:
1821:
1818:
1813:
1809:
1805:
1802:
1799:
1796:
1791:
1787:
1762:
1757:
1753:
1749:
1746:
1743:
1738:
1734:
1730:
1725:
1721:
1717:
1692:
1675:
1671:
1667:
1664:
1661:
1656:
1652:
1648:
1643:
1639:
1619:
1616:
1607:computation).
1595:
1592:
1538:(for Python).
1527:
1524:
1474:Iliffe vectors
1469:
1466:
1383:
1380:
1350:
1349:
1336:
1332:
1321:
1316:
1312:
1311:
1298:
1294:
1283:
1278:
1274:
1273:
1260:
1256:
1245:
1240:
1236:
1235:
1222:
1218:
1207:
1202:
1198:
1197:
1184:
1180:
1169:
1164:
1160:
1159:
1146:
1142:
1131:
1126:
1122:
1121:
1108:
1105:
1102:
1098:
1085:
1078:
1056:
1055:
1042:
1038:
1027:
1022:
1018:
1017:
1004:
1000:
989:
984:
980:
979:
966:
962:
951:
946:
942:
941:
928:
924:
913:
908:
904:
903:
890:
886:
875:
870:
866:
865:
852:
848:
837:
832:
828:
827:
814:
811:
808:
804:
791:
784:
762:
761:
747:
743:
731:
717:
713:
701:
697:
696:
682:
678:
666:
652:
648:
636:
632:
631:
617:
613:
601:
587:
583:
571:
567:
566:
552:
548:
536:
522:
518:
506:
502:
501:
487:
483:
471:
457:
453:
441:
437:
436:
422:
418:
406:
392:
388:
376:
372:
371:
368:
365:
356:
355:
342:
333:
329:
324:
319:
315:
310:
305:
301:
296:
295:
289:
285:
280:
275:
271:
266:
261:
257:
252:
251:
249:
244:
239:
236:
233:
229:
225:
222:
183:instead of 1.
166:
163:
160:
156:
135:
99:
96:
70:respectively.
38:In computing,
26:
9:
6:
4:
3:
2:
3138:
3127:
3124:
3123:
3121:
3111:
3109:
3104:
3103:
3084:
3078:
3063:
3059:
3053:
3038:
3032:
3017:
3011:
3004:
2994:
2990:
2984:
2969:
2965:
2959:
2944:
2938:
2923:
2919:
2913:
2898:
2892:
2877:
2873:
2867:
2851:
2845:
2830:
2826:
2820:
2813:
2802:
2798:
2792:
2785:
2781:
2776:
2769:on 2003-05-18
2765:
2758:
2757:
2752:
2748:
2743:
2738:
2723:
2719:
2713:
2706:
2700:
2698:
2682:
2678:
2672:
2670:
2662:
2646:
2640:
2633:
2625:
2619:
2612:
2597:
2591:
2576:
2572:
2566:
2551:
2547:
2541:
2527:
2523:
2517:
2513:
2502:
2499:
2496:
2493:
2490:
2487:
2485:
2482:
2479:
2476:
2474:
2471:
2469:
2466:
2465:
2459:
2456:
2452:
2447:
2444:
2440:
2436:
2433:in dimension
2432:
2427:
2413:
2410:
2405:
2401:
2395:
2390:
2387:
2384:
2381:
2378:
2374:
2370:
2365:
2361:
2355:
2350:
2347:
2344:
2340:
2331:
2327:
2326:empty product
2309:
2305:
2300:
2294:
2290:
2284:
2281:
2278:
2273:
2270:
2267:
2263:
2258:
2252:
2247:
2244:
2241:
2237:
2233:
2224:
2216:
2212:
2206:
2203:
2200:
2196:
2192:
2189:
2183:
2178:
2174:
2170:
2165:
2161:
2154:
2149:
2145:
2141:
2136:
2132:
2125:
2120:
2116:
2112:
2107:
2103:
2094:
2089:
2074:
2070:
2065:
2059:
2055:
2049:
2044:
2041:
2038:
2035:
2032:
2028:
2023:
2017:
2012:
2009:
2006:
2002:
1998:
1989:
1981:
1977:
1971:
1967:
1963:
1960:
1954:
1949:
1946:
1943:
1939:
1935:
1930:
1927:
1924:
1920:
1913:
1908:
1905:
1902:
1898:
1894:
1889:
1886:
1883:
1879:
1872:
1867:
1863:
1859:
1854:
1850:
1841:
1836:
1819:
1816:
1811:
1807:
1803:
1800:
1794:
1789:
1785:
1776:
1755:
1751:
1747:
1744:
1741:
1736:
1732:
1728:
1723:
1719:
1708:
1704:
1700:
1695:
1691:
1673:
1669:
1665:
1662:
1659:
1654:
1650:
1646:
1641:
1637:
1629:-dimensional
1628:
1623:
1615:
1613:
1608:
1605:
1601:
1594:Transposition
1591:
1589:
1585:
1579:
1567:
1563:
1559:
1555:
1550:
1548:
1544:
1539:
1537:
1532:
1523:
1521:
1516:
1514:
1510:
1507:, and in the
1506:
1501:
1499:
1495:
1491:
1487:
1483:
1479:
1475:
1465:
1463:
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1431:
1427:
1423:
1418:
1416:
1412:
1408:
1404:
1400:
1396:
1392:
1387:
1379:
1375:
1373:
1334:
1330:
1322:
1317:
1314:
1313:
1296:
1292:
1284:
1279:
1276:
1275:
1258:
1254:
1246:
1241:
1238:
1237:
1220:
1216:
1208:
1203:
1200:
1199:
1182:
1178:
1170:
1165:
1162:
1161:
1144:
1140:
1132:
1127:
1124:
1123:
1106:
1103:
1100:
1096:
1086:
1079:
1076:y + N_y*(x-1)
1072:
1071:
1065:
1063:
1040:
1036:
1028:
1023:
1020:
1019:
1002:
998:
990:
985:
982:
981:
964:
960:
952:
947:
944:
943:
926:
922:
914:
909:
906:
905:
888:
884:
876:
871:
868:
867:
850:
846:
838:
833:
830:
829:
812:
809:
806:
802:
792:
785:
778:
777:
771:
769:
745:
741:
732:
715:
711:
702:
699:
698:
680:
676:
667:
650:
646:
637:
634:
633:
615:
611:
602:
585:
581:
572:
569:
568:
550:
546:
537:
520:
516:
507:
504:
503:
485:
481:
472:
455:
451:
442:
439:
438:
420:
416:
407:
390:
386:
377:
374:
373:
369:
366:
363:
362:
359:
340:
331:
327:
317:
313:
303:
299:
287:
283:
273:
269:
259:
255:
247:
242:
237:
234:
231:
227:
223:
220:
213:
212:
211:
208:
206:
202:
198:
193:
189:
184:
182:
164:
161:
158:
154:
133:
125:
121:
117:
112:
110:
107:and the last
106:
95:
93:
89:
85:
81:
77:
71:
69:
65:
61:
55:
53:
49:
45:
41:
32:
19:
3106:
3086:. Retrieved
3077:
3065:. Retrieved
3061:
3052:
3040:. Retrieved
3031:
3019:. Retrieved
3010:
3002:
2996:. Retrieved
2992:
2983:
2971:. Retrieved
2967:
2958:
2946:. Retrieved
2937:
2925:. Retrieved
2921:
2912:
2900:. Retrieved
2891:
2879:. Retrieved
2875:
2866:
2854:. Retrieved
2844:
2832:. Retrieved
2829:rasdaman.org
2828:
2819:
2811:
2804:. Retrieved
2800:
2791:
2779:
2775:
2764:the original
2755:
2737:
2725:. Retrieved
2721:
2712:
2684:. Retrieved
2680:
2659:
2652:. Retrieved
2639:
2630:
2618:
2610:
2603:. Retrieved
2590:
2578:. Retrieved
2574:
2565:
2553:. Retrieved
2549:
2540:
2529:. Retrieved
2525:
2516:
2489:Morton order
2478:Index origin
2450:
2448:
2442:
2438:
2434:
2428:
2092:
2090:
1839:
1837:
1774:
1702:
1698:
1693:
1689:
1626:
1624:
1621:
1609:
1597:
1586:
1551:
1540:
1533:
1529:
1517:
1502:
1471:
1419:
1388:
1385:
1376:
1371:
1353:
1059:
765:
357:
209:
200:
196:
191:
187:
185:
146:, the entry
123:
119:
113:
108:
104:
101:
72:
56:
43:
39:
37:
3021:12 November
2927:12 November
2902:18 November
2856:13 February
2806:25 November
2801:Scilab Wiki
2751:Best, Nicky
2654:18 November
2605:13 November
2555:19 November
2455:permutation
1399:Objective-C
76:CPU caching
68:AoS and SoA
3088:2015-05-16
3042:6 February
2998:2017-11-23
2948:6 February
2727:9 November
2580:2 February
2531:2021-04-10
2508:References
2495:CSR format
2324:where the
1434:GNU Octave
1366:, and the
2968:SciPy.org
2406:ℓ
2379:ℓ
2375:∏
2366:ℓ
2345:ℓ
2341:∏
2295:ℓ
2282:−
2268:ℓ
2264:∏
2238:∑
2225:⋯
2204:−
2190:⋯
2184:⋅
2155:⋅
2126:⋅
2060:ℓ
2033:ℓ
2029:∏
2003:∑
1990:⋯
1961:⋯
1955:⋅
1947:−
1928:−
1914:⋅
1906:−
1887:−
1873:⋅
1817:−
1795:∈
1745:…
1666:×
1663:⋯
1660:×
1647:×
1558:OpenGL ES
1547:Armadillo
1411:Speakeasy
782:x + N_x*y
201:rightmost
3120:Category
3058:"Tensor"
3037:"Tensor"
2852:. Oracle
2462:See also
2332:, i.e.,
1462:Rasdaman
364:Address
197:leftmost
116:matrices
3100:Sources
2973:3 April
2922:Wolfram
2686:31 July
2661:order).
1562:IRIS GL
1422:Fortran
1372:because
1073:Address
1062:Fortran
779:Address
3126:Arrays
2834:29 May
2431:stride
1625:For a
1554:OpenGL
1505:Python
1496:, and
1460:, and
1458:Yorick
1454:Scilab
1446:S-PLUS
1430:MATLAB
1413:, and
1407:Pascal
1360:A(i,j)
1319:A(2,3)
1281:A(1,3)
1243:A(2,2)
1205:A(1,2)
1167:A(2,1)
1129:A(1,1)
1083:A(y,x)
1080:Access
786:Access
192:second
124:column
60:matrix
3067:8 May
2881:1 May
2767:(PDF)
2760:(PDF)
2722:Julia
2648:(PDF)
2627:(PDF)
2601:. IBM
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