Developer Reference for Intel® oneAPI Math Kernel Library for C
p?latrd
Reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal/unitary similarity transformation.
Syntax
voidpslatrd ( char*uplo , MKL_INT*n , MKL_INT*nb , float*a , MKL_INT*ia , MKL_INT*ja , MKL_INT*desca , float*d , float*e , float*tau , float*w , MKL_INT*iw , MKL_INT*jw , MKL_INT*descw , float*work );
voidpdlatrd ( char*uplo , MKL_INT*n , MKL_INT*nb , double*a , MKL_INT*ia , MKL_INT*ja , MKL_INT*desca , double*d , double*e , double*tau , double*w , MKL_INT*iw , MKL_INT*jw , MKL_INT*descw , double*work );
voidpclatrd ( char*uplo , MKL_INT*n , MKL_INT*nb , MKL_Complex8*a , MKL_INT*ia , MKL_INT*ja , MKL_INT*desca , float*d , float*e , MKL_Complex8*tau , MKL_Complex8*w , MKL_INT*iw , MKL_INT*jw , MKL_INT*descw , MKL_Complex8*work );
voidpzlatrd ( char*uplo , MKL_INT*n , MKL_INT*nb , MKL_Complex16*a , MKL_INT*ia , MKL_INT*ja , MKL_INT*desca , double*d , double*e , MKL_Complex16*tau , MKL_Complex16*w , MKL_INT*iw , MKL_INT*jw , MKL_INT*descw , MKL_Complex16*work );
Include Files
mkl_scalapack.h
Description
pslatrd pdlatrd pclatrd pzlatrd The p?latrd routine function reduces nb rows and columns of a real symmetric or complex Hermitian matrix sub( A )= A(ia:ia+n-1 , ja:ja+n-1) to symmetric/complex tridiagonal form by an orthogonal/unitary similarity transformation Q'*sub(A)*Q , and returns the matrices V and W , which are needed to apply the transformation to the unreduced part of sub( A ).
If uplo = U , p?latrd reduces the last nb rows and columns of a matrix, of which the upper triangle is supplied;
if uplo = L , p?latrd reduces the first nb rows and columns of a matrix, of which the lower triangle is supplied.
This is an auxiliary routine function called by p?sytrd (Reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation.) / p?hetrd (Reduces a Hermitian matrix to Hermitian tridiagonal form by a unitary similarity transformation.) .
Input Parameters
uplo
(global)
Specifies whether the upper or lower triangular part of the symmetric/Hermitian matrix sub( A ) is stored: = ‘U’ : Upper triangular = L : Lower triangular.
n
(global)
The number of rows and columns to be operated on, that is, the order of the distributed matrix sub( A ). n ≥ 0 .
nb
(global)
The number of rows and columns to be reduced.
- a
-
REAL for pslatrd DOUBLE PRECISION for pdlatrd COMPLEX for pclatrd COMPLEX*16 for pzlatrd .
Pointer into the local memory to an array of size (lld_a, LOCc(ja+n-1))lld_a * LOCc ( ja + n -1) .
On entry, this array contains the local pieces of the symmetric/Hermitian distributed matrix sub( A ). If uplo = U , the leading n -by- n upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If uplo = L , the leading n -by- n lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced.
ia
(global)
The row index in the global matrix A indicating the first row of sub( A ).
ja
(global)
The column index in the global matrix A indicating the first column of sub( A ).
desca
(global and local) array of size dlen_ . The array descriptor for the distributed matrix A .
iw
(global)
The row index in the global matrix W indicating the first row of sub( W ).
jw
(global)
The column index in the global matrix W indicating the first column of sub( W ).
descw
(global and local) array of size dlen_ . The array descriptor for the distributed matrix W .
- work
-
(local) REAL for pslatrd DOUBLE PRECISION for pdlatrd COMPLEX for pclatrd COMPLEX*16 for pzlatrd . Workspace array of size nb_a .
Output Parameters
- a
-
(local) On exit, if uplo = 'U' , the last nb columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements above the diagonal with the array tau represent the orthogonal/unitary matrix Q as a product of elementary reflectors; if uplo = ‘L’ , the first nb columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements below the diagonal with the array tau represent the orthogonal/unitary matrix Q as a product of elementary reflectors.
- d
-
(local) REAL for pslatrd/pclatrd DOUBLE PRECISION for pdlatrd/pzlatrd . Array of size LOCc(ja+n-1) .
The diagonal elements of the tridiagonal matrix T : d(i) = a(i,i)d [ i ] = A (i+1,i+1), i = 0, 1, …, LOCc(ja+n-1)-1 . d is tied to the distributed matrix A .
- e
-
(local) REAL for pslatrd/pclatrd DOUBLE PRECISION for pdlatrd/pzlatrd . Array of size LOCc(ja+n-1) if uplo = 'U', LOCc(ja+n-2) otherwise. The off-diagonal elements of the tridiagonal matrix T : e(i) = a(i, i + 1) if uplo = 'U' , e(i) = a(i + 1, i) if uplo = 'L' . e[i] =A ( i + 1, i + 2) if uplo = 'U' , e[i] =A ( i + 2, i + 1) if uplo = 'L' , i = 0, 1, …, LOCc(ja+n-1)-1 . e is tied to the distributed matrix A .
- tau
-
(local) REAL for pslatrd DOUBLE PRECISION for pdlatrd COMPLEX for pclatrd COMPLEX*16 for pzlatrd . Array of size LOCc(ja+n-1) . This array contains the scalar factors of the elementary reflectors. tau is tied to the distributed matrix A .
- w
-
(local) REAL for pslatrd DOUBLE PRECISION for pdlatrd COMPLEX for pclatrd COMPLEX*16 for pzlatrd .
Pointer into the local memory to an array of size lld_w * nb_w . This array contains the local pieces of the n -by- nb_w matrix w required to update the unreduced part of sub( A ).
Application Notes
Q = H(n)*H(n-1)*...*H(n-nb+1)H(i) = I - tau*v*v' ,where tau is a real/complex scalar, and v is a real/complex vector with v(i:n) = 0 and v(i-1) = 1 ; v(1:i-1) is stored on exit in A(ia:ia+i-1 , ja+i) , and tau in tau(ja+i-1)[ja+i-2] .
Q = H(1)*H(2)*...*H(nb)H(i) = I - tau*v*v' ,where tau is a real/complex scalar, and v is a real/complex vector with v(1:i) = 0 and v(i+1) = 1 ; v(i+2: n ) is stored on exit in A(ia+i+1: ia+n-1, ja+i-1) , and tau in tau(ja+i-1)[ja+i-2] .
The elements of the vectors v together form the n -by- nb matrix V which is needed, with W , to apply the transformation to the unreduced part of the matrix, using a symmetric/Hermitian rank-2 k update of the form:
sub(A) := sub(A)-vw'-wv' .
The contents of a on exit are illustrated by the following examples with
n = 5 and nb = 2 :
where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H (i).