Matrix Arguments
Matrix arguments of the Intel® Math Kernel Library routines can be stored in either one or twodimensional arrays, using the following storage schemes:

conventional full storage (in a twodimensional array)

packed storage for Hermitian, symmetric, or triangular matrices (in a onedimensional array)

band storage for band matrices (in a twodimensional array)

rectangular full packed storage for symmetric, Hermitian, or triangular matrices as compact as the Packed storage while maintaining efficiency by using Level 3 BLAS/LAPACK kernels.
Full storage is the following obvious scheme: a matrix A is stored in a twodimensional array a, with the matrix element a_{ij}
stored in the array element a(i,j)
.
If a matrix is triangular (upper or lower, as specified by the argument uplo), only the elements of the relevant triangle are stored; the remaining elements of the array need not be set.
Routines that handle symmetric or Hermitian matrices allow for either the upper or lower triangle of the matrix to be stored in the corresponding elements of the array:
 if
uplo ='U'
, 
a_{ij}
is stored ina(i,j)
fori ≤ j
, other elements of a need not be set.  if
uplo ='L',

a_{ij}
is stored ina(i,j)
forj ≤ i
, other elements of a need not be set.
Packed storage allows you to store symmetric, Hermitian, or triangular matrices more compactly: the relevant triangle (again, as specified by the argument uplo) is packed by columns in a onedimensional array ap:
if uplo ='U', a_{ij}
is stored in ap(i+j(j1)/2)
for i ≤ j
if uplo ='L'
, a_{ij}
is stored in ap(i+(2*nj)*(j1)/2)
for j ≤ i
.
In descriptions of LAPACK routines, arrays with packed matrices have names ending in p.
Band storage is as follows: an mbyn band matrix with kl nonzero subdiagonals and ku nonzero superdiagonals is stored compactly in a twodimensional array ab with kl+ku+1 rows and n columns. Columns of the matrix are stored in the corresponding columns of the array, and diagonals of the matrix are stored in rows of the array. Thus,
a_{ij}
is stored in ab(ku+1+ij,j) for max(1,jku) ≤ i ≤ min(n,j+kl).
Use the band storage scheme only when kl and ku are much less than the matrix size n. Although the routines work correctly for all values of kl and ku, using the band storage is inefficient if your matrices are not really banded.
The band storage scheme is illustrated by the following example, when
m = n = 6, kl = 2, ku = 1
Array elements marked * are not used by the routines:
When a general band matrix is supplied for LU factorization, space must be allowed to store kl additional superdiagonals generated by fillin as a result of row interchanges. This means that the matrix is stored according to the above scheme, but with kl + ku
superdiagonals. Thus,
a_{ij}
is stored in ab(kl+ku+1+ij,j)
for max(1,jku) ≤ i ≤ min(n,j+kl)
.
The band storage scheme for LU factorization is illustrated by the following example, whenm = n = 6, kl = 2, ku = 1:
Array elements marked * are not used by the routines; elements marked + need not be set on entry, but are required by the LU factorization routines to store the results. The input array will be overwritten on exit by the details of the LU factorization as follows:
where u_{ij}
are the elements of the upper triangular matrix U, and m_{ij}
are the multipliers used during factorization.
Triangular band matrices are stored in the same format, with either kl= 0 if upper triangular, or ku = 0
if lower triangular. For symmetric or Hermitian band matrices with k subdiagonals or superdiagonals, you need to store only the upper or lower triangle, as specified by the argument uplo:
if uplo ='U', a_{ij}
is stored in ab(k+1+ij,j) for max(1,jk) ≤ i ≤ j
if uplo ='L', a_{ij}
is stored in ab(1+ij,j) for j ≤ i ≤ min(n,j+k).
In descriptions of LAPACK routines, arrays that hold matrices in band storage have names ending in b.
In Fortran, columnmajor ordering of storage is assumed. This means that elements of the same column occupy successive storage locations.
Three quantities are usually associated with a twodimensional array argument: its leading dimension, which specifies the number of storage locations between elements in the same row, its number of rows, and its number of columns. For a matrix in full storage, the leading dimension of the array must be at least as large as the number of rows in the matrix.
A character transposition parameter is often passed to indicate whether the matrix argument is to be used in normal or transposed form or, for a complex matrix, if the conjugate transpose of the matrix is to be used.
The values of the transposition parameter for these three cases are the following:
 'N' or 'n'

normal (no conjugation, no transposition)
 'T' or 't'

transpose
 'C' or 'c'

conjugate transpose.
Example. TwoDimensional Complex Array
Suppose A (1:5, 1:4)
is the complex twodimensional array presented by matrix
Let transa be the transposition parameter, m be the number of rows, n be the number of columns, and lda be the leading dimension. Then if
transa = 'N'
, m = 4
, n = 2
, and lda = 5
, the matrix argument would be
If transa = 'T'
, m = 4
, n = 2
, and lda =5
, the matrix argument would be
If transa = 'C'
, m = 4
, n = 2
, and lda =5
, the matrix argument would be
Note that care should be taken when using a leading dimension value which is different from the number of rows specified in the declaration of the twodimensional array. For example, suppose the array A above is declared as COMPLEX A (5,4)
.
Then if transa = 'N', m = 3, n = 4
, and lda = 4
, the matrix argument will be
Rectangular Full Packed storage allows you to store symmetric, Hermitian, or triangular matrices as compact as the Packed storage while maintaining efficiency by using Level 3 BLAS/LAPACK kernels. To store an nbyn triangle (and suppose for simplicity that n is even), you partition the triangle into three parts: two n/2byn/2 triangles and an n/2byn/2 square, then pack this as an nbyn/2 rectangle (or n/2byn rectangle), by transposing (or transposeconjugating) one of the triangles and packing it next to the other triangle. Since the two triangles are stored in full storage, you can use existing efficient routines on them.
There are eight cases of RFP storage representation: when n is even or odd, the packed matrix is transposed or not, the triangular matrix is lower or upper. See below for all the eight storage schemes illustrated:
n is odd, A is lower triangular
n is even, A is lower triangular
n is odd, A is upper triangular
n is even, A is upper triangular
Intel MKL provides a number of routines such as ?hfrk, ?sfrk performing BLAS operations working directly on RFP matrices, as well as some conversion routines, for instance, ?tpttf goes from the standard packed format to RFP and ?trttf goes from the full format to RFP.
Please refer to the Netlib site for more information.
Note that in the descriptions of LAPACK routines, arrays with RFP matrices have names ending in fp.