Developer Reference for Intel® oneAPI Math Kernel Library for C
?sbtrd
Reduces a real symmetric band matrix to tridiagonal form.
Syntax
lapack_intLAPACKE_ssbtrd ( intmatrix_layout , charvect , charuplo , lapack_intn , lapack_intkd , float*ab , lapack_intldab , float*d , float*e , float*q , lapack_intldq );
lapack_intLAPACKE_dsbtrd ( intmatrix_layout , charvect , charuplo , lapack_intn , lapack_intkd , double*ab , lapack_intldab , double*d , double*e , double*q , lapack_intldq );
Include Files
mkl.h
Description
ssbtrd dsbtrd sbtrd
The routine reduces a real symmetric band matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A = Q*T*Q^{T} . The orthogonal matrix Q is determined as a product of Givens rotations.
If required, the routine can also form the matrix Q explicitly.
Input Parameters
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
vect
Must be ‘V’ , ‘N’ , or ‘U’ .
If vect = 'V' , the routine returns the explicit matrix Q . If vect = 'N' , the routine does not return Q . If vect = 'U' , the routine updates matrix X by forming X * Q .
uplo
Must be ‘U’ or ‘L’ .
If uplo = 'U' , ab stores the upper triangular part of A . If uplo = 'L' , ab stores the lower triangular part of A .
n
The order of the matrix A ( n≥0 ).
kd
The number of super- or sub-diagonals in A
( kd≥0 ).
- ab , q , work
-
REAL for ssbtrd DOUBLE PRECISION for dsbtrd .
ab (size at least max(1, ldab * n ) for column major layout and at least max(1, ldab *( kd + 1)) for row major layout) is an array containing either upper or lower triangular part of the matrix A (as specified by uplo ) in band storage format.
The second dimension of ab must be at least max(1, n ).
q (size max(1, ldq * n )) is an array.
If vect = 'U' , the q array must contain an n -by- n matrix X . If vect = 'N' or 'V' , the q parameter need not be set. The second dimension of q must be at least max(1, n ). work (*) is a workspace array. The dimension of work must be at least max(1, n ).
ldab
The leading dimension of ab ; at least kd +1 for column major layout and n for row major layout .
ldq
The leading dimension of q . Constraints:
ldq≥ max(1, n) if vect = 'V' or 'U' ; ldq≥ 1 if vect = 'N' .
Output Parameters
- ab
-
On exit, the diagonal elements of the array ab are overwritten by the diagonal elements of the tridiagonal matrix T . If kd > 0, the elements on the first superdiagonal (if uplo = 'U' ) or the first subdiagonal (if uplo = 'L' ) are ovewritten by the off-diagonal elements of T . The rest of ab is overwritten by values generated during the reduction.
- d , e , q
-
REAL for ssbtrd DOUBLE PRECISION for dsbtrd . Arrays:
d contains the diagonal elements of the matrix T .
The size of d must be at least max(1, n ).
e contains the off-diagonal elements of T .
The size of e must be at least max(1, n -1).
q is not referenced if vect = 'N' .
If vect = 'V' , q contains the n -by- n matrix Q . If vect = 'U' , q contains the product X * Q . The second dimension of q must be: at least max(1, n ) if vect = 'V' ; at least 1 if vect = 'N' .
Return Values
This function returns a value info .
If info=0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
Application Notes
The computed matrix \(T\) is exactly similar to a matrix \(A\) + \(E\) , where \(||E||_{2} = c(n) \varepsilon ||A||_{2}\) , c(n) is a modestly increasing function of n , and \(\varepsilon\) is the machine precision. The computed matrix \(Q\) differs from an exactly orthogonal matrix by a matrix \(E\) such that \(||E||_{2} = O(\varepsilon)\) .
The total number of floating-point operations is approximately \(6n^{2} kd\) if vect = 'N' , with \(3n^{3} (kd-1)/kd\) additional operations if vect = 'V' .
The complex counterpart of this routine is hbtrd .