LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ sgeqr2()

 subroutine sgeqr2 ( integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer info )

SGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.

Download SGEQR2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` SGEQR2 computes a QR factorization of a real m-by-n matrix A:

A = Q * ( R ),
( 0 )

where:

Q is a m-by-m orthogonal matrix;
R is an upper-triangular n-by-n matrix;
0 is a (m-n)-by-n zero matrix, if m > n.```
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, the elements on and above the diagonal of the array contain the min(m,n) by n upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, 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(1) H(2) . . . H(k), 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(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).```

Definition at line 129 of file sgeqr2.f.

130*
131* -- LAPACK computational routine --
132* -- LAPACK is a software package provided by Univ. of Tennessee, --
133* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
134*
135* .. Scalar Arguments ..
136 INTEGER INFO, LDA, M, N
137* ..
138* .. Array Arguments ..
139 REAL A( LDA, * ), TAU( * ), WORK( * )
140* ..
141*
142* =====================================================================
143*
144* .. Parameters ..
145 REAL ONE
146 parameter( one = 1.0e+0 )
147* ..
148* .. Local Scalars ..
149 INTEGER I, K
150 REAL AII
151* ..
152* .. External Subroutines ..
153 EXTERNAL slarf, slarfg, xerbla
154* ..
155* .. Intrinsic Functions ..
156 INTRINSIC max, min
157* ..
158* .. Executable Statements ..
159*
160* Test the input arguments
161*
162 info = 0
163 IF( m.LT.0 ) THEN
164 info = -1
165 ELSE IF( n.LT.0 ) THEN
166 info = -2
167 ELSE IF( lda.LT.max( 1, m ) ) THEN
168 info = -4
169 END IF
170 IF( info.NE.0 ) THEN
171 CALL xerbla( 'SGEQR2', -info )
172 RETURN
173 END IF
174*
175 k = min( m, n )
176*
177 DO 10 i = 1, k
178*
179* Generate elementary reflector H(i) to annihilate A(i+1:m,i)
180*
181 CALL slarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1,
182 \$ tau( i ) )
183 IF( i.LT.n ) THEN
184*
185* Apply H(i) to A(i:m,i+1:n) from the left
186*
187 aii = a( i, i )
188 a( i, i ) = one
189 CALL slarf( 'Left', m-i+1, n-i, a( i, i ), 1, tau( i ),
190 \$ a( i, i+1 ), lda, work )
191 a( i, i ) = aii
192 END IF
193 10 CONTINUE
194 RETURN
195*
196* End of SGEQR2
197*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
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
subroutine slarfg(n, alpha, x, incx, tau)
SLARFG generates an elementary reflector (Householder matrix).
Definition slarfg.f:106
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