/* Subroutine */ int sgeqr2_(integer *m, integer *n, real *a, integer *lda,
real *tau, real *work, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
/* Local variables */
integer i__, k;
real aii;
extern /* Subroutine */ int slarf_(char *, integer *, integer *, real *,
integer *, real *, real *, integer *, real *), xerbla_(
char *, integer *), slarfp_(integer *, real *, real *,
integer *, real *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGEQR2 computes a QR factorization of a real m by n matrix A: */
/* A = 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) 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). */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,M). */
/* TAU (output) REAL array, dimension (min(M,N)) */
/* The scalar factors of the elementary reflectors (see Further */
/* Details). */
/* WORK (workspace) REAL array, dimension (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(k), where k = min(m,n). */
/* Each H(i) has the form */
/* H(i) = 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), */
/* and tau in TAU(i). */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input arguments */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--tau;
--work;
/* Function Body */
*info = 0;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*m)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGEQR2", &i__1);
return 0;
}
k = min(*m,*n);
i__1 = k;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Generate elementary reflector H(i) to annihilate A(i+1:m,i) */
i__2 = *m - i__ + 1;
/* Computing MIN */
i__3 = i__ + 1;
slarfp_(&i__2, &a[i__ + i__ * a_dim1], &a[min(i__3, *m)+ i__ * a_dim1]
, &c__1, &tau[i__]);
if (i__ < *n) {
/* Apply H(i) to A(i:m,i+1:n) from the left */
aii = a[i__ + i__ * a_dim1];
a[i__ + i__ * a_dim1] = 1.f;
i__2 = *m - i__ + 1;
i__3 = *n - i__;
slarf_("Left", &i__2, &i__3, &a[i__ + i__ * a_dim1], &c__1, &tau[
i__], &a[i__ + (i__ + 1) * a_dim1], lda, &work[1]);
a[i__ + i__ * a_dim1] = aii;
}
/* L10: */
}
return 0;
/* End of SGEQR2 */
} /* sgeqr2_ */
int slaqps_(int *m, int *n, int *offset, int
*nb, int *kb, float *a, int *lda, int *jpvt, float *tau,
float *vn1, float *vn2, float *auxv, float *f, int *ldf)
{
/* System generated locals */
int a_dim1, a_offset, f_dim1, f_offset, i__1, i__2;
float r__1, r__2;
/* Builtin functions */
double sqrt(double);
int i_nint(float *);
/* Local variables */
int j, k, rk;
float akk;
int pvt;
float temp, temp2;
extern double snrm2_(int *, float *, int *);
float tol3z;
extern int sgemm_(char *, char *, int *, int *,
int *, float *, float *, int *, float *, int *, float *,
float *, int *);
int itemp;
extern int sgemv_(char *, int *, int *, float *,
float *, int *, float *, int *, float *, float *, int *), sswap_(int *, float *, int *, float *, int *);
extern double slamch_(char *);
int lsticc;
extern int isamax_(int *, float *, int *);
extern int slarfp_(int *, float *, float *, int *,
float *);
int lastrk;
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLAQPS computes a step of QR factorization with column pivoting */
/* of a float M-by-N matrix A by using Blas-3. It tries to factorize */
/* NB columns from A starting from the row OFFSET+1, and updates all */
/* of the matrix with Blas-3 xGEMM. */
/* In some cases, due to catastrophic cancellations, it cannot */
/* factorize NB columns. Hence, the actual number of factorized */
/* columns is returned in KB. */
/* Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized. */
/* 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 */
/* OFFSET (input) INTEGER */
/* The number of rows of A that have been factorized in */
/* previous steps. */
/* NB (input) INTEGER */
/* The number of columns to factorize. */
/* KB (output) INTEGER */
/* The number of columns actually factorized. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the M-by-N matrix A. */
/* On exit, block A(OFFSET+1:M,1:KB) is the triangular */
/* factor obtained and block A(1:OFFSET,1:N) has been */
/* accordingly pivoted, but no factorized. */
/* The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has */
/* been updated. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= MAX(1,M). */
/* JPVT (input/output) INTEGER array, dimension (N) */
/* JPVT(I) = K <==> Column K of the full matrix A has been */
/* permuted into position I in AP. */
/* TAU (output) REAL array, dimension (KB) */
/* The scalar factors of the elementary reflectors. */
/* VN1 (input/output) REAL array, dimension (N) */
/* The vector with the partial column norms. */
/* VN2 (input/output) REAL array, dimension (N) */
/* The vector with the exact column norms. */
/* AUXV (input/output) REAL array, dimension (NB) */
/* Auxiliar vector. */
/* F (input/output) REAL array, dimension (LDF,NB) */
/* Matrix F' = L*Y'*A. */
/* LDF (input) INTEGER */
/* The leading dimension of the array F. LDF >= MAX(1,N). */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain */
/* X. Sun, Computer Science Dept., Duke University, USA */
/* Partial column norm updating strategy modified by */
/* Z. Drmac and Z. Bujanovic, Dept. of Mathematics, */
/* University of Zagreb, Croatia. */
/* June 2006. */
/* For more details see LAPACK Working Note 176. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--jpvt;
--tau;
--vn1;
--vn2;
--auxv;
f_dim1 = *ldf;
f_offset = 1 + f_dim1;
f -= f_offset;
/* Function Body */
/* Computing MIN */
i__1 = *m, i__2 = *n + *offset;
lastrk = MIN(i__1,i__2);
lsticc = 0;
k = 0;
tol3z = sqrt(slamch_("Epsilon"));
/* Beginning of while loop. */
L10:
if (k < *nb && lsticc == 0) {
++k;
rk = *offset + k;
/* Determine ith pivot column and swap if necessary */
i__1 = *n - k + 1;
pvt = k - 1 + isamax_(&i__1, &vn1[k], &c__1);
if (pvt != k) {
sswap_(m, &a[pvt * a_dim1 + 1], &c__1, &a[k * a_dim1 + 1], &c__1);
i__1 = k - 1;
sswap_(&i__1, &f[pvt + f_dim1], ldf, &f[k + f_dim1], ldf);
itemp = jpvt[pvt];
jpvt[pvt] = jpvt[k];
jpvt[k] = itemp;
vn1[pvt] = vn1[k];
vn2[pvt] = vn2[k];
}
/* Apply previous Householder reflectors to column K: */
/* A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)'. */
if (k > 1) {
i__1 = *m - rk + 1;
i__2 = k - 1;
sgemv_("No transpose", &i__1, &i__2, &c_b8, &a[rk + a_dim1], lda,
&f[k + f_dim1], ldf, &c_b9, &a[rk + k * a_dim1], &c__1);
}
/* Generate elementary reflector H(k). */
if (rk < *m) {
i__1 = *m - rk + 1;
slarfp_(&i__1, &a[rk + k * a_dim1], &a[rk + 1 + k * a_dim1], &
c__1, &tau[k]);
} else {
slarfp_(&c__1, &a[rk + k * a_dim1], &a[rk + k * a_dim1], &c__1, &
tau[k]);
}
akk = a[rk + k * a_dim1];
a[rk + k * a_dim1] = 1.f;
/* Compute Kth column of F: */
/* Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)'*A(RK:M,K). */
if (k < *n) {
i__1 = *m - rk + 1;
i__2 = *n - k;
sgemv_("Transpose", &i__1, &i__2, &tau[k], &a[rk + (k + 1) *
a_dim1], lda, &a[rk + k * a_dim1], &c__1, &c_b16, &f[k +
1 + k * f_dim1], &c__1);
}
/* Padding F(1:K,K) with zeros. */
i__1 = k;
for (j = 1; j <= i__1; ++j) {
f[j + k * f_dim1] = 0.f;
/* L20: */
}
/* Incremental updating of F: */
/* F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)' */
/* *A(RK:M,K). */
if (k > 1) {
i__1 = *m - rk + 1;
i__2 = k - 1;
r__1 = -tau[k];
sgemv_("Transpose", &i__1, &i__2, &r__1, &a[rk + a_dim1], lda, &a[
rk + k * a_dim1], &c__1, &c_b16, &auxv[1], &c__1);
i__1 = k - 1;
sgemv_("No transpose", n, &i__1, &c_b9, &f[f_dim1 + 1], ldf, &
auxv[1], &c__1, &c_b9, &f[k * f_dim1 + 1], &c__1);
}
/* Update the current row of A: */
/* A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)'. */
if (k < *n) {
i__1 = *n - k;
sgemv_("No transpose", &i__1, &k, &c_b8, &f[k + 1 + f_dim1], ldf,
&a[rk + a_dim1], lda, &c_b9, &a[rk + (k + 1) * a_dim1],
lda);
}
/* Update partial column norms. */
if (rk < lastrk) {
i__1 = *n;
for (j = k + 1; j <= i__1; ++j) {
if (vn1[j] != 0.f) {
/* NOTE: The following 4 lines follow from the analysis in */
/* Lapack Working Note 176. */
temp = (r__1 = a[rk + j * a_dim1], ABS(r__1)) / vn1[j];
/* Computing MAX */
r__1 = 0.f, r__2 = (temp + 1.f) * (1.f - temp);
temp = MAX(r__1,r__2);
/* Computing 2nd power */
r__1 = vn1[j] / vn2[j];
temp2 = temp * (r__1 * r__1);
if (temp2 <= tol3z) {
vn2[j] = (float) lsticc;
lsticc = j;
} else {
vn1[j] *= sqrt(temp);
}
}
/* L30: */
}
}
a[rk + k * a_dim1] = akk;
/* End of while loop. */
goto L10;
}
*kb = k;
rk = *offset + *kb;
/* Apply the block reflector to the rest of the matrix: */
/* A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) - */
/* A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)'. */
/* Computing MIN */
i__1 = *n, i__2 = *m - *offset;
if (*kb < MIN(i__1,i__2)) {
i__1 = *m - rk;
i__2 = *n - *kb;
sgemm_("No transpose", "Transpose", &i__1, &i__2, kb, &c_b8, &a[rk +
1 + a_dim1], lda, &f[*kb + 1 + f_dim1], ldf, &c_b9, &a[rk + 1
+ (*kb + 1) * a_dim1], lda);
}
/* Recomputation of difficult columns. */
L40:
if (lsticc > 0) {
itemp = i_nint(&vn2[lsticc]);
i__1 = *m - rk;
vn1[lsticc] = snrm2_(&i__1, &a[rk + 1 + lsticc * a_dim1], &c__1);
/* NOTE: The computation of VN1( LSTICC ) relies on the fact that */
/* SNRM2 does not fail on vectors with norm below the value of */
/* SQRT(DLAMCH('S')) */
vn2[lsticc] = vn1[lsticc];
lsticc = itemp;
goto L40;
}
return 0;
/* End of SLAQPS */
} /* slaqps_ */
int stzrqf_(int *m, int *n, float *a, int *lda,
float *tau, int *info)
{
/* System generated locals */
int a_dim1, a_offset, i__1, i__2;
float r__1;
/* Local variables */
int i__, k, m1;
extern int sger_(int *, int *, float *, float *,
int *, float *, int *, float *, int *), sgemv_(char *,
int *, int *, float *, float *, int *, float *, int *
, float *, float *, int *), scopy_(int *, float *,
int *, float *, int *), saxpy_(int *, float *, float *,
int *, float *, int *), xerbla_(char *, int *),
slarfp_(int *, float *, float *, int *, float *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* This routine is deprecated and has been replaced by routine STZRZF. */
/* STZRQF reduces the M-by-N ( M<=N ) float upper trapezoidal matrix A */
/* to upper triangular form by means of orthogonal transformations. */
/* The upper trapezoidal matrix A is factored as */
/* A = ( R 0 ) * Z, */
/* where Z is an N-by-N orthogonal matrix and R is an M-by-M upper */
/* triangular matrix. */
/* 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 >= M. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the leading M-by-N upper trapezoidal part of the */
/* array A must contain the matrix to be factorized. */
/* On exit, the leading M-by-M upper triangular part of A */
/* contains the upper triangular matrix R, and elements M+1 to */
/* N of the first M rows of A, with the array TAU, represent the */
/* orthogonal matrix Z as a product of M elementary reflectors. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= MAX(1,M). */
/* TAU (output) REAL array, dimension (M) */
/* The scalar factors of the elementary reflectors. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Further Details */
/* =============== */
/* The factorization is obtained by Householder's method. The kth */
/* transformation matrix, Z( k ), which is used to introduce zeros into */
/* the ( m - k + 1 )th row of A, is given in the form */
/* Z( k ) = ( I 0 ), */
/* ( 0 T( k ) ) */
/* where */
/* T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), */
/* ( 0 ) */
/* ( z( k ) ) */
/* tau is a scalar and z( k ) is an ( n - m ) element vector. */
/* tau and z( k ) are chosen to annihilate the elements of the kth row */
/* of X. */
/* The scalar tau is returned in the kth element of TAU and the vector */
/* u( k ) in the kth row of A, such that the elements of z( k ) are */
/* in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in */
/* the upper triangular part of A. */
/* Z is given by */
/* Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--tau;
/* Function Body */
*info = 0;
if (*m < 0) {
*info = -1;
} else if (*n < *m) {
*info = -2;
} else if (*lda < MAX(1,*m)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("STZRQF", &i__1);
return 0;
}
/* Perform the factorization. */
if (*m == 0) {
return 0;
}
if (*m == *n) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
tau[i__] = 0.f;
/* L10: */
}
} else {
/* Computing MIN */
i__1 = *m + 1;
m1 = MIN(i__1,*n);
for (k = *m; k >= 1; --k) {
/* Use a Householder reflection to zero the kth row of A. */
/* First set up the reflection. */
i__1 = *n - *m + 1;
slarfp_(&i__1, &a[k + k * a_dim1], &a[k + m1 * a_dim1], lda, &tau[
k]);
if (tau[k] != 0.f && k > 1) {
/* We now perform the operation A := A*P( k ). */
/* Use the first ( k - 1 ) elements of TAU to store a( k ), */
/* where a( k ) consists of the first ( k - 1 ) elements of */
/* the kth column of A. Also let B denote the first */
/* ( k - 1 ) rows of the last ( n - m ) columns of A. */
i__1 = k - 1;
scopy_(&i__1, &a[k * a_dim1 + 1], &c__1, &tau[1], &c__1);
/* Form w = a( k ) + B*z( k ) in TAU. */
i__1 = k - 1;
i__2 = *n - *m;
sgemv_("No transpose", &i__1, &i__2, &c_b8, &a[m1 * a_dim1 +
1], lda, &a[k + m1 * a_dim1], lda, &c_b8, &tau[1], &
c__1);
/* Now form a( k ) := a( k ) - tau*w */
/* and B := B - tau*w*z( k )'. */
i__1 = k - 1;
r__1 = -tau[k];
saxpy_(&i__1, &r__1, &tau[1], &c__1, &a[k * a_dim1 + 1], &
c__1);
i__1 = k - 1;
i__2 = *n - *m;
r__1 = -tau[k];
sger_(&i__1, &i__2, &r__1, &tau[1], &c__1, &a[k + m1 * a_dim1]
, lda, &a[m1 * a_dim1 + 1], lda);
}
/* L20: */
}
}
return 0;
/* End of STZRQF */
} /* stzrqf_ */
int slaqp2_(int *m, int *n, int *offset, float *a,
int *lda, int *jpvt, float *tau, float *vn1, float *vn2, float *
work)
{
/* System generated locals */
int a_dim1, a_offset, i__1, i__2, i__3;
float r__1, r__2;
/* Builtin functions */
double sqrt(double);
/* Local variables */
int i__, j, mn;
float aii;
int pvt;
float temp, temp2;
extern double snrm2_(int *, float *, int *);
float tol3z;
int offpi;
extern int slarf_(char *, int *, int *, float *,
int *, float *, float *, int *, float *);
int itemp;
extern int sswap_(int *, float *, int *, float *,
int *);
extern double slamch_(char *);
extern int isamax_(int *, float *, int *);
extern int slarfp_(int *, float *, float *, int *,
float *);
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLAQP2 computes a QR factorization with column pivoting of */
/* the block A(OFFSET+1:M,1:N). */
/* The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized. */
/* 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. */
/* OFFSET (input) INTEGER */
/* The number of rows of the matrix A that must be pivoted */
/* but no factorized. OFFSET >= 0. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the M-by-N matrix A. */
/* On exit, the upper triangle of block A(OFFSET+1:M,1:N) is */
/* the triangular factor obtained; the elements in block */
/* A(OFFSET+1:M,1:N) below the diagonal, together with the */
/* array TAU, represent the orthogonal matrix Q as a product of */
/* elementary reflectors. Block A(1:OFFSET,1:N) has been */
/* accordingly pivoted, but no factorized. */
/* 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) REAL array, dimension (MIN(M,N)) */
/* The scalar factors of the elementary reflectors. */
/* VN1 (input/output) REAL array, dimension (N) */
/* The vector with the partial column norms. */
/* VN2 (input/output) REAL array, dimension (N) */
/* The vector with the exact column norms. */
/* WORK (workspace) REAL array, dimension (N) */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain */
/* X. Sun, Computer Science Dept., Duke University, USA */
/* Partial column norm updating strategy modified by */
/* Z. Drmac and Z. Bujanovic, Dept. of Mathematics, */
/* University of Zagreb, Croatia. */
/* June 2006. */
/* For more details see LAPACK Working Note 176. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--jpvt;
--tau;
--vn1;
--vn2;
--work;
/* Function Body */
/* Computing MIN */
i__1 = *m - *offset;
mn = MIN(i__1,*n);
tol3z = sqrt(slamch_("Epsilon"));
/* Compute factorization. */
i__1 = mn;
for (i__ = 1; i__ <= i__1; ++i__) {
offpi = *offset + i__;
/* Determine ith pivot column and swap if necessary. */
i__2 = *n - i__ + 1;
pvt = i__ - 1 + isamax_(&i__2, &vn1[i__], &c__1);
if (pvt != i__) {
sswap_(m, &a[pvt * a_dim1 + 1], &c__1, &a[i__ * a_dim1 + 1], &
c__1);
itemp = jpvt[pvt];
jpvt[pvt] = jpvt[i__];
jpvt[i__] = itemp;
vn1[pvt] = vn1[i__];
vn2[pvt] = vn2[i__];
}
/* Generate elementary reflector H(i). */
if (offpi < *m) {
i__2 = *m - offpi + 1;
slarfp_(&i__2, &a[offpi + i__ * a_dim1], &a[offpi + 1 + i__ *
a_dim1], &c__1, &tau[i__]);
} else {
slarfp_(&c__1, &a[*m + i__ * a_dim1], &a[*m + i__ * a_dim1], &
c__1, &tau[i__]);
}
if (i__ < *n) {
/* Apply H(i)' to A(offset+i:m,i+1:n) from the left. */
aii = a[offpi + i__ * a_dim1];
a[offpi + i__ * a_dim1] = 1.f;
i__2 = *m - offpi + 1;
i__3 = *n - i__;
slarf_("Left", &i__2, &i__3, &a[offpi + i__ * a_dim1], &c__1, &
tau[i__], &a[offpi + (i__ + 1) * a_dim1], lda, &work[1]);
a[offpi + i__ * a_dim1] = aii;
}
/* Update partial column norms. */
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
if (vn1[j] != 0.f) {
/* NOTE: The following 4 lines follow from the analysis in */
/* Lapack Working Note 176. */
/* Computing 2nd power */
r__2 = (r__1 = a[offpi + j * a_dim1], ABS(r__1)) / vn1[j];
temp = 1.f - r__2 * r__2;
temp = MAX(temp,0.f);
/* Computing 2nd power */
r__1 = vn1[j] / vn2[j];
temp2 = temp * (r__1 * r__1);
if (temp2 <= tol3z) {
if (offpi < *m) {
i__3 = *m - offpi;
vn1[j] = snrm2_(&i__3, &a[offpi + 1 + j * a_dim1], &
c__1);
vn2[j] = vn1[j];
} else {
vn1[j] = 0.f;
vn2[j] = 0.f;
}
} else {
vn1[j] *= sqrt(temp);
}
}
/* L10: */
}
/* L20: */
}
return 0;
/* End of SLAQP2 */
} /* slaqp2_ */
/* Subroutine */ int sgerq2_(integer *m, integer *n, real *a, integer *lda,
real *tau, real *work, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2;
/* Local variables */
integer i__, k;
real aii;
/* -- LAPACK routine (version 3.2) -- */
/* November 2006 */
/* Purpose */
/* ======= */
/* SGERQ2 computes an RQ factorization of a real m by n matrix A: */
/* A = R * Q. */
/* 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) REAL array, dimension (LDA,N) */
/* On entry, the m by n matrix A. */
/* On exit, if m <= n, the upper triangle of the subarray */
/* A(1:m,n-m+1:n) contains the m by m upper triangular matrix R; */
/* if m >= n, the elements on and above the (m-n)-th subdiagonal */
/* contain the m by n upper trapezoidal matrix R; the remaining */
/* elements, with the array TAU, represent the orthogonal matrix */
/* Q as a product of elementary reflectors (see Further */
/* Details). */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,M). */
/* TAU (output) REAL array, dimension (min(M,N)) */
/* The scalar factors of the elementary reflectors (see Further */
/* Details). */
/* WORK (workspace) REAL array, dimension (M) */
/* 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(k), where k = min(m,n). */
/* Each H(i) has the form */
/* H(i) = I - tau * v * v' */
/* where tau is a real scalar, and v is a real vector with */
/* v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in */
/* A(m-k+i,1:n-k+i-1), and tau in TAU(i). */
/* ===================================================================== */
/* Test the input arguments */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--tau;
--work;
/* Function Body */
*info = 0;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*m)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGERQ2", &i__1);
return 0;
}
k = min(*m,*n);
for (i__ = k; i__ >= 1; --i__) {
/* Generate elementary reflector H(i) to annihilate */
/* A(m-k+i,1:n-k+i-1) */
i__1 = *n - k + i__;
slarfp_(&i__1, &a[*m - k + i__ + (*n - k + i__) * a_dim1], &a[*m - k
+ i__ + a_dim1], lda, &tau[i__]);
/* Apply H(i) to A(1:m-k+i-1,1:n-k+i) from the right */
aii = a[*m - k + i__ + (*n - k + i__) * a_dim1];
a[*m - k + i__ + (*n - k + i__) * a_dim1] = 1.f;
i__1 = *m - k + i__ - 1;
i__2 = *n - k + i__;
slarf_("Right", &i__1, &i__2, &a[*m - k + i__ + a_dim1], lda, &tau[
i__], &a[a_offset], lda, &work[1]);
a[*m - k + i__ + (*n - k + i__) * a_dim1] = aii;
}
return 0;
/* End of SGERQ2 */
} /* sgerq2_ */
/* Subroutine */ int sgeqpf_(integer *m, integer *n, real *a, integer *lda,
integer *jpvt, real *tau, real *work, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j, ma, mn;
real aii;
integer pvt;
real temp, temp2;
extern doublereal snrm2_(integer *, real *, integer *);
real tol3z;
extern /* Subroutine */ int slarf_(char *, integer *, integer *, real *,
integer *, real *, real *, integer *, real *);
integer itemp;
extern /* Subroutine */ int sswap_(integer *, real *, integer *, real *,
integer *), sgeqr2_(integer *, integer *, real *, integer *, real
*, real *, integer *), sorm2r_(char *, char *, integer *, integer
*, integer *, real *, integer *, real *, real *, integer *, real *
, integer *);
extern doublereal slamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
extern integer isamax_(integer *, real *, integer *);
extern /* Subroutine */ int slarfp_(integer *, real *, real *, integer *,
real *);
/* -- LAPACK deprecated driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* This routine is deprecated and has been replaced by routine SGEQP3. */
/* SGEQPF 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) REAL 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) REAL array, dimension (min(M,N)) */
/* The scalar factors of the elementary reflectors. */
/* WORK (workspace) REAL 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. */
/* Partial column norm updating strategy modified by */
/* Z. Drmac and Z. Bujanovic, Dept. of Mathematics, */
/* University of Zagreb, Croatia. */
/* June 2006. */
/* For more details see LAPACK Working Note 176. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input arguments */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--jpvt;
--tau;
--work;
/* Function Body */
*info = 0;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*m)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGEQPF", &i__1);
return 0;
}
mn = min(*m,*n);
tol3z = sqrt(slamch_("Epsilon"));
/* Move initial columns up front */
itemp = 1;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (jpvt[i__] != 0) {
if (i__ != itemp) {
sswap_(m, &a[i__ * a_dim1 + 1], &c__1, &a[itemp * a_dim1 + 1],
&c__1);
jpvt[i__] = jpvt[itemp];
jpvt[itemp] = i__;
} else {
jpvt[i__] = i__;
}
++itemp;
} else {
jpvt[i__] = i__;
}
/* L10: */
}
--itemp;
/* Compute the QR factorization and update remaining columns */
if (itemp > 0) {
ma = min(itemp,*m);
sgeqr2_(m, &ma, &a[a_offset], lda, &tau[1], &work[1], info);
if (ma < *n) {
i__1 = *n - ma;
sorm2r_("Left", "Transpose", m, &i__1, &ma, &a[a_offset], lda, &
tau[1], &a[(ma + 1) * a_dim1 + 1], lda, &work[1], info);
}
}
if (itemp < mn) {
/* Initialize partial column norms. The first n elements of */
/* work store the exact column norms. */
i__1 = *n;
for (i__ = itemp + 1; i__ <= i__1; ++i__) {
i__2 = *m - itemp;
work[i__] = snrm2_(&i__2, &a[itemp + 1 + i__ * a_dim1], &c__1);
work[*n + i__] = work[i__];
/* L20: */
}
/* Compute factorization */
i__1 = mn;
for (i__ = itemp + 1; i__ <= i__1; ++i__) {
/* Determine ith pivot column and swap if necessary */
i__2 = *n - i__ + 1;
pvt = i__ - 1 + isamax_(&i__2, &work[i__], &c__1);
if (pvt != i__) {
sswap_(m, &a[pvt * a_dim1 + 1], &c__1, &a[i__ * a_dim1 + 1], &
c__1);
itemp = jpvt[pvt];
jpvt[pvt] = jpvt[i__];
jpvt[i__] = itemp;
work[pvt] = work[i__];
work[*n + pvt] = work[*n + i__];
}
/* Generate elementary reflector H(i) */
if (i__ < *m) {
i__2 = *m - i__ + 1;
slarfp_(&i__2, &a[i__ + i__ * a_dim1], &a[i__ + 1 + i__ *
a_dim1], &c__1, &tau[i__]);
} else {
slarfp_(&c__1, &a[*m + *m * a_dim1], &a[*m + *m * a_dim1], &
c__1, &tau[*m]);
}
if (i__ < *n) {
/* Apply H(i) to A(i:m,i+1:n) from the left */
aii = a[i__ + i__ * a_dim1];
a[i__ + i__ * a_dim1] = 1.f;
i__2 = *m - i__ + 1;
i__3 = *n - i__;
slarf_("LEFT", &i__2, &i__3, &a[i__ + i__ * a_dim1], &c__1, &
tau[i__], &a[i__ + (i__ + 1) * a_dim1], lda, &work[(*
n << 1) + 1]);
a[i__ + i__ * a_dim1] = aii;
}
/* Update partial column norms */
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
if (work[j] != 0.f) {
/* NOTE: The following 4 lines follow from the analysis in */
/* Lapack Working Note 176. */
temp = (r__1 = a[i__ + j * a_dim1], dabs(r__1)) / work[j];
/* Computing MAX */
r__1 = 0.f, r__2 = (temp + 1.f) * (1.f - temp);
temp = dmax(r__1,r__2);
/* Computing 2nd power */
r__1 = work[j] / work[*n + j];
temp2 = temp * (r__1 * r__1);
if (temp2 <= tol3z) {
if (*m - i__ > 0) {
i__3 = *m - i__;
work[j] = snrm2_(&i__3, &a[i__ + 1 + j * a_dim1],
&c__1);
work[*n + j] = work[j];
} else {
work[j] = 0.f;
work[*n + j] = 0.f;
}
} else {
work[j] *= sqrt(temp);
}
}
/* L30: */
}
/* L40: */
}
}
return 0;
/* End of SGEQPF */
} /* sgeqpf_ */
