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* ======================================================================
SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
*
* -- LAPACK test routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* March 31, 1993
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DGEQPF computes a QR factorization with column pivoting of a
* real M-by-N matrix A: A*P = Q*R.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, the upper triangle of the array contains the
* min(M,N)-by-N upper triangular matrix R; the elements
* below the diagonal, together with the array TAU,
* represent the orthogonal matrix Q as a product of
* min(m,n) elementary reflectors.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* JPVT (input/output) INTEGER array, dimension (N)
* On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
* to the front of A*P (a leading column); if JPVT(i) = 0,
* the i-th column of A is a free column.
* On exit, if JPVT(i) = k, then the i-th column of A*P
* was the k-th column of A.
*
* TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
* The scalar factors of the elementary reflectors.
*
* WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
*
* INFO (output) 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(n)
*
* Each H(i) has the form
*
* H = I - tau * v * v'
*
* 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).
*
* The matrix P is represented in jpvt as follows: If
* jpvt(j) = i
* then the jth column of P is the ith canonical unit vector.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, ITEMP, J, MA, MN, PVT
DOUBLE PRECISION AII, TEMP, TEMP2
* ..
* .. External Subroutines ..
EXTERNAL DGEQR2, DLARF, DLARFG, DORM2R, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DNRM2
EXTERNAL IDAMAX, DNRM2
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQPF', -INFO )
RETURN
END IF
*
MN = MIN( M, N )
*
* Move initial columns up front
*
ITEMP = 1
DO 10 I = 1, N
IF( JPVT( I ).NE.0 ) THEN
IF( I.NE.ITEMP ) THEN
CALL DSWAP( M, A( 1, I ), 1, A( 1, ITEMP ), 1 )
JPVT( I ) = JPVT( ITEMP )
JPVT( ITEMP ) = I
ELSE
JPVT( I ) = I
END IF
ITEMP = ITEMP + 1
ELSE
JPVT( I ) = I
END IF
10 CONTINUE
ITEMP = ITEMP - 1
*
* Compute the QR factorization and update remaining columns
*
IF( ITEMP.GT.0 ) THEN
MA = MIN( ITEMP, M )
CALL DGEQR2( M, MA, A, LDA, TAU, WORK, INFO )
IF( MA.LT.N ) THEN
CALL DORM2R( 'Left', 'Transpose', M, N-MA, MA, A, LDA, TAU,
$ A( 1, MA+1 ), LDA, WORK, INFO )
END IF
END IF
*
IF( ITEMP.LT.MN ) THEN
*
* Initialize partial column norms. The first n elements of
* work store the exact column norms.
*
DO 20 I = ITEMP + 1, N
WORK( I ) = DNRM2( M-ITEMP, A( ITEMP+1, I ), 1 )
WORK( N+I ) = WORK( I )
20 CONTINUE
*
* Compute factorization
*
DO 40 I = ITEMP + 1, MN
*
* Determine ith pivot column and swap if necessary
*
PVT = ( I-1 ) + IDAMAX( N-I+1, WORK( I ), 1 )
*
IF( PVT.NE.I ) THEN
CALL DSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 )
ITEMP = JPVT( PVT )
JPVT( PVT ) = JPVT( I )
JPVT( I ) = ITEMP
WORK( PVT ) = WORK( I )
WORK( N+PVT ) = WORK( N+I )
END IF
*
* Generate elementary reflector H(i)
*
IF( I.LT.M ) THEN
CALL DLARFG( M-I+1, A( I, I ), A( I+1, I ), 1, TAU( I ) )
ELSE
CALL DLARFG( 1, A( M, M ), A( M, M ), 1, TAU( M ) )
END IF
*
IF( I.LT.N ) THEN
*
* Apply H(i) to A(i:m,i+1:n) from the left
*
AII = A( I, I )
A( I, I ) = ONE
CALL DLARF( 'LEFT', M-I+1, N-I, A( I, I ), 1, TAU( I ),
$ A( I, I+1 ), LDA, WORK( 2*N+1 ) )
A( I, I ) = AII
END IF
*
* Update partial column norms
*
DO 30 J = I + 1, N
IF( WORK( J ).NE.ZERO ) THEN
TEMP = ONE - ( ABS( A( I, J ) ) / WORK( J ) )**2
TEMP = MAX( TEMP, ZERO )
TEMP2 = ONE + 0.05D0*TEMP*
$ ( WORK( J ) / WORK( N+J ) )**2
IF( TEMP2.EQ.ONE ) THEN
IF( M-I.GT.0 ) THEN
WORK( J ) = DNRM2( M-I, A( I+1, J ), 1 )
WORK( N+J ) = WORK( J )
ELSE
WORK( J ) = ZERO
WORK( N+J ) = ZERO
END IF
ELSE
WORK( J ) = WORK( J )*SQRT( TEMP )
END IF
END IF
30 CONTINUE
*
40 CONTINUE
END IF
RETURN
*
* End of DGEQPF
*
END
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