Developer Reference for Intel® oneAPI Math Kernel Library for C
?sptrd
Reduces a real symmetric matrix to tridiagonal form using packed storage.
Syntax
lapack_intLAPACKE_ssptrd ( intmatrix_layout , charuplo , lapack_intn , float*ap , float*d , float*e , float*tau );
lapack_intLAPACKE_dsptrd ( intmatrix_layout , charuplo , lapack_intn , double*ap , double*d , double*e , double*tau );
Include Files
mkl.h
Description
ssptrd dsptrd sptrd
The routine reduces a packed 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 for details.
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' , ap stores the packed upper triangle of A . If uplo = 'L' , ap stores the packed lower triangle of A .
n
The order of the matrix A ( n≥ 0 ).
- ap
-
REAL for ssptrd DOUBLE PRECISION for dsptrd .
Array, size at least max(1, n(n+1)/2). Contains either upper or lower triangle of A (as specified by uplo ) in the packed form described in Matrix Storage Schemes .
Output Parameters
- ap
-
Overwritten by the tridiagonal matrix T and details of the orthogonal matrix Q , as specified by uplo .
- d , e , tau
-
REAL for ssptrd DOUBLE PRECISION for dsptrd . Arrays:
d contains the diagonal elements of the matrix T .
The dimension of d must be at least max(1, n ).
e contains the off-diagonal elements of T .
The dimension of e must be at least max(1, n -1).
tau Stores ( n -1) scalars that define elementary reflectors in decomposition of the matrix Q in a product of n -1 reflectors.
The dimension of tau must be at least max(1, n -1).
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 matrix \(Q\) is represented as a product of n -1 elementary reflectors , as follows :
If uplo = ‘U’ , \(Q = H(n-1) \cdots H(2) H(1)\)
Each \(H\) (i) has the form
\(H(i) = I - \tau v v^{T}\)
where tau is a real scalar and v is a real vector with v(i+1:n) = 0 and v(i) = 1 .
On exit, tau is stored in tau [ i - 1] , and v(1:i-1) is stored in AP , overwriting A(1:i-1, i+1) .
If uplo = ‘L’ , \(Q = H(1) H(2) \cdots H(n-1)\)
Each \(H\) (i) has the form
\(H(i) = I - \tau v v^{T}\)
where tau is a real scalar and v is a real vector with v(1:i) = 0 and v(i+1) = 1 .
On exit, tau is stored in tau [ i - 1] , and v(i+2:n) is stored in AP , overwriting A(i+2:n, i) .
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 hptrd .