Developer Reference for Intel® oneAPI Math Kernel Library for C
?sytrd
Reduces a real symmetric matrix to tridiagonal form.
Syntax
lapack_intLAPACKE_ssytrd ( intmatrix_layout , charuplo , lapack_intn , float*a , lapack_intlda , float*d , float*e , float*tau );
lapack_intLAPACKE_dsytrd ( intmatrix_layout , charuplo , lapack_intn , double*a , lapack_intlda , double*d , double*e , double*tau );
Include Files
mkl.h
Description
ssytrd dsytrd sytrd
The routine reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A = Q*T*Q^{T} . The orthogonal matrix Q is not formed explicitly but is represented as a product of n -1 elementary reflectors. Routines are provided for working with Q in this representation (see Application Notes below) .
Input Parameters
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
uplo
Must be ‘U’ or ‘L’ .
If uplo = 'U' , a stores the upper triangular part of A . If uplo = 'L' , a stores the lower triangular part of A .
n
The order of the matrix A ( n≥ 0 ).
- a , work
-
REAL for ssytrd DOUBLE PRECISION for dsytrd .
a (size max(1, lda * n )) is an array containing either upper or lower triangular part of the matrix A , as specified by uplo . If uplo = 'U' , the leading n -by- n upper triangular part of a contains the upper triangular part of the matrix A , and the strictly lower triangular part of A is not referenced. If uplo = 'L' , the leading n -by- n lower triangular part of a contains the lower triangular part of the matrix A , and the strictly upper triangular part of A is not referenced.
The second dimension of a must be at least max(1, n) . work is a workspace array, its dimension max(1, lwork) .
lda
The leading dimension of a ; at least max(1, n ).
lwork
The size of the work array ( lwork≥n ).
If lwork = -1 , then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla . See Application Notes for the suggested value of lwork .
Output Parameters
- a
-
On exit, if uplo = 'U' , the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T , and the elements above the first superdiagonal, with the array tau , represent the orthogonal matrix Q as a product of elementary reflectors; if uplo = 'L' , the diagonal and first subdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T , and the elements below the first subdiagonal, with the array tau , represent the orthogonal matrix Q as a product of elementary reflectors.
- d , e , tau
-
REAL for ssytrd DOUBLE PRECISION for dsytrd . 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).
tau stores ( n -1) scalars that define elementary reflectors in decomposition of the orthogonal matrix Q in a product of n-1 elementary reflectors. tau(n) is used as workspace.
The size of tau must be at least max(1, n ).
- work(1)
-
If info=0 , on exit work(1) contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.
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 approximate number of floating-point operations is \((4/3)n^{3}\) .
After calling this routine, you can call the following:
to form the computed matrix \(Q\) explicitly
to multiply a real matrix by \(Q\) .
The complex counterpart of this routine is ?hetrd .