Contents

# ?gelsd

Computes the minimum-norm solution to a linear least squares problem using the singular value decomposition of A and a divide and conquer method.

## Syntax

Include Files
• mkl.h
Description
The routine computes the minimum-norm solution to a real linear least squares problem:
minimize ||
b
-
A
*
x
||
2
using the singular value decomposition (SVD) 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 problem is solved in three steps:
1. Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a "bidiagonal least squares problem" (BLS).
2. Solve the BLS using a divide and conquer approach.
3. Apply back all the Householder transformations to solve the original least squares problem.
The effective rank of
A
is determined by treating as zero those singular values which are less than
rcond
times the largest singular value.
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
.
rcond
rcond
is used to determine the effective rank of
A
. Singular values
s
(
i
)
rcond
*
s
(1)
are treated as zero. If
rcond
0
, machine precision is used instead.
Output Parameters
a
On exit,
A
has been overwritten.
b
Overwritten by the
n
-by-
nrhs
solution matrix
X
.
If
m
n
and
rank
=
n
, the residual sum-of-squares for the solution in the
i
-th column is given by the sum of squares of modulus of elements
n
+1:
m
in that column.
s
Array, size at least max(1, min(
m
,
n
)). The singular values of
A
in decreasing order. The condition number of
A
in the 2-norm is
k
2
(
A
) =
s
(1)/
s
(min(
m
,
n
))
.
rank
The effective rank of
A
, that is, the number of singular values which are greater than
rcond
*
s
(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.
If
info
=
i
, then the algorithm for computing the SVD failed to converge;
i
indicates the number of off-diagonal elements of an intermediate bidiagonal form that did not converge to zero.

#### Product and Performance Information

1

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