LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ sgeql2()

 subroutine sgeql2 ( integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO )

SGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:
``` SGEQL2 computes a QL factorization of a real m by n matrix A:
A = Q * L.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0.``` [in,out] A ``` A is REAL array, dimension (LDA,N) On entry, the m by n matrix A. On exit, if m >= n, the lower triangle of the subarray A(m-n+1:m,1:n) contains the n by n lower triangular matrix L; if m <= n, the elements on and below the (n-m)-th superdiagonal contain the m by n lower trapezoidal matrix L; the remaining elements, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] TAU ``` TAU is REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details).``` [out] WORK ` WORK is REAL array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details:
```  The matrix Q is represented as a product of elementary reflectors

Q = H(k) . . . H(2) H(1), where k = min(m,n).

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(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
A(1:m-k+i-1,n-k+i), and tau in TAU(i).```

Definition at line 122 of file sgeql2.f.

123 *
124 * -- LAPACK computational routine --
125 * -- LAPACK is a software package provided by Univ. of Tennessee, --
126 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
127 *
128 * .. Scalar Arguments ..
129  INTEGER INFO, LDA, M, N
130 * ..
131 * .. Array Arguments ..
132  REAL A( LDA, * ), TAU( * ), WORK( * )
133 * ..
134 *
135 * =====================================================================
136 *
137 * .. Parameters ..
138  REAL ONE
139  parameter( one = 1.0e+0 )
140 * ..
141 * .. Local Scalars ..
142  INTEGER I, K
143  REAL AII
144 * ..
145 * .. External Subroutines ..
146  EXTERNAL slarf, slarfg, xerbla
147 * ..
148 * .. Intrinsic Functions ..
149  INTRINSIC max, min
150 * ..
151 * .. Executable Statements ..
152 *
153 * Test the input arguments
154 *
155  info = 0
156  IF( m.LT.0 ) THEN
157  info = -1
158  ELSE IF( n.LT.0 ) THEN
159  info = -2
160  ELSE IF( lda.LT.max( 1, m ) ) THEN
161  info = -4
162  END IF
163  IF( info.NE.0 ) THEN
164  CALL xerbla( 'SGEQL2', -info )
165  RETURN
166  END IF
167 *
168  k = min( m, n )
169 *
170  DO 10 i = k, 1, -1
171 *
172 * Generate elementary reflector H(i) to annihilate
173 * A(1:m-k+i-1,n-k+i)
174 *
175  CALL slarfg( m-k+i, a( m-k+i, n-k+i ), a( 1, n-k+i ), 1,
176  \$ tau( i ) )
177 *
178 * Apply H(i) to A(1:m-k+i,1:n-k+i-1) from the left
179 *
180  aii = a( m-k+i, n-k+i )
181  a( m-k+i, n-k+i ) = one
182  CALL slarf( 'Left', m-k+i, n-k+i-1, a( 1, n-k+i ), 1, tau( i ),
183  \$ a, lda, work )
184  a( m-k+i, n-k+i ) = aii
185  10 CONTINUE
186  RETURN
187 *
188 * End of SGEQL2
189 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
subroutine slarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
SLARF applies an elementary reflector to a general rectangular matrix.
Definition: slarf.f:124
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