Index notation

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Index notation is used in mathematics to refer to the elements of matrices or the components of a vector. The formalism of how indices are used varies according to the discipline. In particular, there are different methods for referring to the elements of a list, a vector, or a matrix, depending on whether one is writing a formal mathematical paper for publication, or when one is writing a computer program. In some fields, the terms "index notation", or "indicial notation" are used to refer to Einstein notation.

It is quite common in some mathematical proofs to refer to the elements of an array using subscripts, and in some cases superscripts. The use of superscripts is frequently encountered in the theory of general relativity. The following line states in effect that the each of the elements of a vector c are equal to the sum of the corresponding elements of vectors a and b

$\mathbf{c}_j = \mathbf{a}_j + \mathbf{b}_j$

$\mathbf{c}_0 = \mathbf{a}_0 + \mathbf{b}_0$

$\mathbf{c}_1 = \mathbf{a}_1 + \mathbf{b}_1$

and so on.

Superscripts are used instead of subscripts to distinguish covariant from contravariant entities, see Covariance and contravariance.

[edit] Index notation in computing

In several programming languages, index notation is a way of addressing elements of an array. This method is used since it is closest to how it is implemented in assembly language whereby the address of the first element is used as a base, and a multiple (the index) of the element size is used to address inside the array.

For example, if an array of integers is stored in a region of the computer's memory starting at the memory cell with address 3000 (the base address), and each integer occupies four cells (bytes), then the elements of this array are at memory locations 3000, 3004, 3008, ..., 0x3000 + 4(n-1). In general, the address of the ith element of an array with base address b and element size s is b+is.

[edit] C implementation details

In the C programming language, we can write the above as *(base + i) (pointer form) or base[i] (array indexing form), which is exactly equivalent because the C standard defines the array indexing form as a transformation to pointer form. Coincidentally, since pointer addition is commutative, this allows for obscure expressions such as 3[base] which is equivalent to base[3].

[edit] Multidimensional arrays

Things become more interesting when we consider arrays with more than one index, for example, a two-dimensional table. We have three possibilities:

make the two-dimensional array one-dimensional by computing a single index from the two
consider a one-dimensional array where each element is another one-dimensional array, i.e an array of arrays
use additional storage to hold the array of addresses of each row of the origninal array, and store the rows of the original array as separate one-dimensional arrays

In C, all three methods can be used. When the first method is used, the programmer decides how the elements of the array are laid out in the computer's memory, and provides the formulas to compute the location of each element. The second method is used when the number of elements in each row is the same and known at the time the program is written. The programmer declares the array to have, say, three columns by writing e.g. elementtype tablename[][3];. One then refers to a particular element of the array by writing tablename[first index][second index]. The compiler computes the total number of memory cells occupied by each row, uses the first index to find the address of the desired row, and then uses the second index to find the address of the desired element in the row. When the third method is used, the programmer declares the table to be an array of pointers, like in elementtype *tablename[];. When the programmer subsequently specifies a particular element tablename[first index][second index], the compiler generates instructions to look up the address of the row specified by the first index, and use this address as the base when computing the address of the element specified by the second index.

[edit] Example

This function multiplies two 3x3 floating point matrices together.

void mult3x3f(float result[][3], const float A[][3], const float B[][3])
{
  int i, j, k;
  for (i = 0; i < 3; ++i) {
    for (j = 0; j < 3; ++j) {
      result[i][j] = 0;
      for (k = 0; k < 3; ++k)
        result[i][j] += A[i][k] * B[k][j];
    }
  }
}

[edit] In other languages

In other programming languages such as Pascal, indices may start at 1, so indexing in a block of memory can be changed to fit a start-at-1 addressing scheme by a simple linear transformation - in this scheme, the memory location of the ith element with base address b and element size s is b+(i-1)s.