In computing, row-major order and column-major order describe methods for storing multidimensional arrays in linear memory. Following standard matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
notation, rows are numbered by the first index of a two-dimensional array and columns by the second index. Array layout is critical for correctly passing arrays between programs written in different languages. It is also important for performance when traversing an array because accessing array elements that are contiguous in memory is usually faster than accessing elements which are not, due to caching
Cache
In computer engineering, a cache is a component that transparently stores data so that future requests for that data can be served faster. The data that is stored within a cache might be values that have been computed earlier or duplicates of original values that are stored elsewhere...
.
Row-major order is used in C,PL/I; column-major order is used in Fortran and MATLAB.
Row-major order
In row-major storage, a multidimensional array in linear memory is accessed such that rows are stored one after the other. It is the approach used by the C programming language
C (programming language)
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
as well as many other languages, with the notable exceptions of Fortran
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
.
When using row-major order, the difference between addresses of array cells in increasing rows is larger than addresses of cells in increasing columns. For example, consider this 2×3 array:
An array declared in C as
int A[2][3] = { {1, 2, 3}, {4, 5, 6} };
would be laid out contiguously in linear memory as:
1 2 3 4 5 6
To traverse this array in the order in which it is laid out in memory, one would use the following nested loop:
for (i = 0; i < 2; i++)
for (j = 0; j < 3; j++)
printf("%d\n", A[i][j]);
The difference in offset from one column to the next is 1 and from one row to the next is 3. The linear offset from the beginning of the array to any given element A[row][column] can then be computed as:
offset = row*NUMCOLS + column
where NUMCOLS is the number of columns in the array.
The above formula only works when using the C convention of labeling the first element 0. In other words, row 1, column 2 in matrix A, would be represented as A[0][1].
Note that this technique generalizes, so a 2×3×4 array looks like:
int A[2][3][4] =
In computing, row-major order and column-major order describe methods for storing multidimensional arrays in linear memory. Following standard matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
notation, rows are numbered by the first index of a two-dimensional array and columns by the second index. Array layout is critical for correctly passing arrays between programs written in different languages. It is also important for performance when traversing an array because accessing array elements that are contiguous in memory is usually faster than accessing elements which are not, due to caching
Cache
In computer engineering, a cache is a component that transparently stores data so that future requests for that data can be served faster. The data that is stored within a cache might be values that have been computed earlier or duplicates of original values that are stored elsewhere...
.
Row-major order is used in C,PL/I; column-major order is used in Fortran and MATLAB.
Row-major order
In row-major storage, a multidimensional array in linear memory is accessed such that rows are stored one after the other. It is the approach used by the C programming language
C (programming language)
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
as well as many other languages, with the notable exceptions of Fortran
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
.
When using row-major order, the difference between addresses of array cells in increasing rows is larger than addresses of cells in increasing columns. For example, consider this 2×3 array:
An array declared in C as
int A[2][3] = { {1, 2, 3}, {4, 5, 6} };
would be laid out contiguously in linear memory as:
1 2 3 4 5 6
To traverse this array in the order in which it is laid out in memory, one would use the following nested loop:
for (i = 0; i < 2; i++)
for (j = 0; j < 3; j++)
printf("%d\n", A[i][j]);
The difference in offset from one column to the next is 1 and from one row to the next is 3. The linear offset from the beginning of the array to any given element A[row][column] can then be computed as:
offset = row*NUMCOLS + column
where NUMCOLS is the number of columns in the array.
The above formula only works when using the C convention of labeling the first element 0. In other words, row 1, column 2 in matrix A, would be represented as A[0][1].
Note that this technique generalizes, so a 2×3×4 array looks like:
int A[2][3][4] =
In computing, row-major order and column-major order describe methods for storing multidimensional arrays in linear memory. Following standard matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
notation, rows are numbered by the first index of a two-dimensional array and columns by the second index. Array layout is critical for correctly passing arrays between programs written in different languages. It is also important for performance when traversing an array because accessing array elements that are contiguous in memory is usually faster than accessing elements which are not, due to caching
Cache
In computer engineering, a cache is a component that transparently stores data so that future requests for that data can be served faster. The data that is stored within a cache might be values that have been computed earlier or duplicates of original values that are stored elsewhere...
.
Row-major order is used in C,PL/I; column-major order is used in Fortran and MATLAB.
Row-major order
In row-major storage, a multidimensional array in linear memory is accessed such that rows are stored one after the other. It is the approach used by the C programming language
C (programming language)
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
as well as many other languages, with the notable exceptions of Fortran
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
.
When using row-major order, the difference between addresses of array cells in increasing rows is larger than addresses of cells in increasing columns. For example, consider this 2×3 array:
An array declared in C as
int A[2][3] = { {1, 2, 3}, {4, 5, 6} };
would be laid out contiguously in linear memory as:
1 2 3 4 5 6
To traverse this array in the order in which it is laid out in memory, one would use the following nested loop:
for (i = 0; i < 2; i++)
for (j = 0; j < 3; j++)
printf("%d\n", A[i][j]);
The difference in offset from one column to the next is 1 and from one row to the next is 3. The linear offset from the beginning of the array to any given element A[row][column] can then be computed as:
offset = row*NUMCOLS + column
where NUMCOLS is the number of columns in the array.
The above formula only works when using the C convention of labeling the first element 0. In other words, row 1, column 2 in matrix A, would be represented as A[0][1].
Note that this technique generalizes, so a 2×3×4 array looks like:
int A[2][3][4] = , };
and the array would be laid out in linear memory as:
Column-major order is a similar method of flattening arrays onto linear memory, but the columns are listed in sequence. The programming languages Fortran
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command-line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB...
R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
In-place matrix transposition, also called in-situ matrix transposition, is the problem of transposing an N×M matrix in-place in computer memory, ideally with O additional storage, or at most with additional storage much less than NM...
, such transpositions are rarely performed explicitly. For example, software libraries for linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
Blas is mainly a Spanish given name and surname, related to Blaise. It may refer to-Places:*Piz Blas, mountain in Switzerland*San Blas , many places - see separate article, also**Cape San Blas Light, lighthouse...
, typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid the necessity of data movement.
Generalization to higher dimensions
It is possible to generalize both of these concepts to arrays with greater than two dimensions. For higher-dimensional arrays, the ordering determines which dimensions of the array are more consecutive in memory. Any of the dimensions could be consecutive, just as a two-dimensional array could be listed column-first or row-first. The difference in offset between listings of that dimension would then be determined by a product of other dimensions. It is uncommon, however, to have any variation except ordering dimensions first to last or last to first. These two variations correspond to row-major and column-major, respectively.
More explicitly, consider a d-dimensional array with dimensions Nk (k=1...d). A given element of this array is specified by a tuple
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
of d (zero-based) indices .
In row-major order, the last dimension is contiguous, so that the memory-offset of this element is given by:
In column-major order, the first dimension is contiguous, so that the memory-offset of this element is given by:
Note that the difference between row-major and column-major order is simply that the order of the dimensions is reversed. Equivalently, in row-major order the rightmost indices vary faster as one steps through consecutive memory locations, while in column-major order the leftmost indices vary faster.
Matrix representation is a method used by a computer language to store matrices of more than one dimension in memory.Fortran and C use different schemes. Fortran uses "Column Major", in which all the elements for a given column are stored contiguously in memory...
In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector...
, the equivalent of turning a matrix into the corresponding column-major vector.
The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.
array with dimensions Nk (k=1...d). A given element of this array is specified by a tuple
of d (zero-based) indices 
.
