Developer Reference for Intel® oneAPI Math Kernel Library for C
?tzrzf
Reduces the upper trapezoidal matrix A to upper triangular form.
Syntax
lapack_intLAPACKE_stzrzf ( intmatrix_layout , lapack_intm , lapack_intn , float*a , lapack_intlda , float*tau );
lapack_intLAPACKE_dtzrzf ( intmatrix_layout , lapack_intm , lapack_intn , double*a , lapack_intlda , double*tau );
lapack_intLAPACKE_ctzrzf ( intmatrix_layout , lapack_intm , lapack_intn , lapack_complex_float*a , lapack_intlda , lapack_complex_float*tau );
lapack_intLAPACKE_ztzrzf ( intmatrix_layout , lapack_intm , lapack_intn , lapack_complex_double*a , lapack_intlda , lapack_complex_double*tau );
Include Files
mkl.h
Description
stzrzf dtzrzf ctzrzf ztzrzf tzrzf
The routine reduces the m -by- n ( m≤n ) real/complex upper trapezoidal matrix A to upper triangular form by means of orthogonal/unitary transformations. The upper trapezoidal matrix A = [ A 1 A 2] = [ A1:m, 1:m , A1:m, m+1:n ] is factored as
A = [R0]*Z ,
where Z is an n -by- n orthogonal/unitary matrix, R is an m -by- m upper triangular matrix, and 0 is the m -by-( n-m ) zero matrix.
See larz that applies an elementary reflector returned by ?tzrzf to a general matrix.
The ?tzrzf routine replaces the deprecated ?tzrqf routine.
Input Parameters
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
m
The number of rows in the matrix A ( m≥ 0 ).
n
The number of columns in A ( n ≥ m ).
- a , work
-
REAL for stzrzf DOUBLE PRECISION for dtzrzf COMPLEX for ctzrzf DOUBLE COMPLEX for ztzrzf . Arrays: a ( lda , *), work ( lwork ). Array a is of size max(1, lda * n ) for column major layout and max(1, lda * m ) for row major layout. The leading m -by- n upper trapezoidal part of the array a contains the matrix A to be factorized. 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, m ) for column major layout and max(1, n ) for row major layout .
lwork
The size of the work array;
lwork≥ max(1, m) . 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
-
Overwritten on exit by the factorization data as follows: the leading m -by- m upper triangular part of a contains the upper triangular matrix R , and elements m +1 to n of the first m rows of a , with the array tau , represent the orthogonal matrix Z as a product of m elementary reflectors.
- tau
-
REAL for stzrzf DOUBLE PRECISION for dtzrzf COMPLEX for ctzrzf DOUBLE COMPLEX for ztzrzf . Array, size at least max (1, m ). Contains scalar factors of the elementary reflectors for the matrix Z .
- 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 factorization is obtained by Householder’s method. The k -th transformation matrix, \(Z(k)\), which is used to introduce zeros into the ( m - k + 1)-th row of \(A\) , is given in the form
where for real flavors
and for complex flavors
tau is a scalar and z(k) is an l -element vector. tau and z(k) are chosen to annihilate the elements of the k- th row of \(A\) 2.
The scalar tau is returned in the k- th element of tau and the vector u(k) in the k- th row of \(A\) , such that the elements of z(k) are stored in a(k, m+1), ..., a(k, n). the last m - n elements of the k -th row of array a .
The elements of \(R\) are returned in the upper triangular part of \(A\) .
The matrix \(Z\) is given by
\(Z = Z(1) Z(2) \cdots Z(m)\) .
Related routines include:
to apply matrix Q (for real matrices)
to apply matrix Q (for complex matrices).