Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

Contents

?gelsy

Computes the minimum-norm solution to a linear least squares problem using a complete orthogonal factorization of A.

Syntax

lapack_int LAPACKE_sgelsy
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
n
,
lapack_int
nrhs
,
float*
a
,
lapack_int
lda
,
float*
b
,
lapack_int
ldb
,
lapack_int*
jpvt
,
float
rcond
,
lapack_int*
rank
);
lapack_int LAPACKE_dgelsy
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
n
,
lapack_int
nrhs
,
double*
a
,
lapack_int
lda
,
double*
b
,
lapack_int
ldb
,
lapack_int*
jpvt
,
double
rcond
,
lapack_int*
rank
);
lapack_int LAPACKE_cgelsy
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
n
,
lapack_int
nrhs
,
lapack_complex_float*
a
,
lapack_int
lda
,
lapack_complex_float*
b
,
lapack_int
ldb
,
lapack_int*
jpvt
,
float
rcond
,
lapack_int*
rank
);
lapack_int LAPACKE_zgelsy
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
n
,
lapack_int
nrhs
,
lapack_complex_double*
a
,
lapack_int
lda
,
lapack_complex_double*
b
,
lapack_int
ldb
,
lapack_int*
jpvt
,
double
rcond
,
lapack_int*
rank
);
Include Files
  • mkl.h
Description
The
?gelsy
 routine computes the minimum-norm solution to a real/complex linear least squares problem:
minimize ||
b
 -
A
*
x
||
2
using a complete orthogonal factorization of
A
.
A
 is an
m
-by-
n
 matrix which may be rank-deficient. Several right hand side vectors
b
 and solution vectors
x
 can be handled in a single call; they are stored as the columns of the
m
-by-
nrhs
 right hand side matrix
B
 and the
n
-by-
nrhs
 solution matrix
X
.
The routine first computes a
QR
 factorization with column pivoting:
Equation
with
R
11
 defined as the largest leading submatrix whose estimated condition number is less than 1/
rcond
. The order of
R
11
,
rank
, is the effective rank of
A
. Then,
R
22
 is considered to be negligible, and
R
12
 is annihilated by orthogonal/unitary transformations from the right, arriving at the complete orthogonal factorization:
Equation
The minimum-norm solution is then
Equation for real flavors and
Equation for complex flavors,
where
Q
1
 consists of the first
rank
 columns of
Q
.
The
?gelsy
 routine is identical to the original deprecated
?gelsx
 routine except for the following differences:
  • The call to the subroutine
    ?geqpf
     has been substituted by the call to the subroutine
    ?geqp3
    , which is a BLAS-3 version of the
    QR
     factorization with column pivoting.
  • The matrix
    B
     (the right hand side) is updated with BLAS-3.
  • The permutation of the matrix
    B
     (the right hand side) is faster and more simple.
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 of the matrix
A
 (
m
≥
 0
).
n
The number of columns of the matrix
A
(
n
≥
 0
).
nrhs
The number of right-hand sides; the number of columns in
B
 (
nrhs
≥
 0
).
a
,
b
Arrays:
a
(size max(1,
lda
*
n
) for column major layout and max(1,
lda
*
m
) for row major layout)
 contains the
m
-by-
n
 matrix
A
.
b
(size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*max(
m
,
n
)) for row major layout)
 contains the
m
-by-
nrhs
 right hand side matrix
B
.
lda
The leading dimension of
a
; at least max(1,
m
)
for column major layout and max(1,
n
) for row major layout
.
ldb
The leading dimension of
b
; must be at least max(1,
m
,
n
)
for column major layout and at least max(1,
nrhs
) for row major layout
.
jpvt
Array, size at least max(1,
n
).
On entry, if
jpvt
[
i
 - 1]
≠
 0, the
i
-th column of
A
 is permuted to the front of
AP
, otherwise the
i
-th column of
A
 is a free column.
rcond
rcond
 is used to determine the effective rank of
A
, which is defined as the order of the largest leading triangular submatrix
R
11
 in the
QR
 factorization with pivoting of
A
, whose estimated condition number < 1/
rcond
.
Output Parameters
a
On exit, overwritten by the details of the complete orthogonal factorization of
A
.
b
Overwritten by the
n
-by-
nrhs
 solution matrix
X
.
jpvt
On exit, if
jpvt
[
i
 - 1]
=
k
, then the
i
-th column of
AP
 was the
k
-th column of
A
.
rank
The effective rank of
A
, that is, the order of the submatrix
R
11
. This is the same as the order of the submatrix
T
11
 in the complete orthogonal factorization of
A
.
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.

Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.