 LAPACK  3.9.1 LAPACK: Linear Algebra PACKage

## ◆ 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.

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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)
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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