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opencv/3rdparty/lapack/dlasyf.c

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#include "clapack.h"
/* Subroutine */ int dlasyf_(char *uplo, integer *n, integer *nb, integer *kb,
doublereal *a, integer *lda, integer *ipiv, doublereal *w, integer *
ldw, integer *info)
{
/* -- LAPACK routine (version 3.1) --
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
Purpose
=======
DLASYF computes a partial factorization of a real symmetric matrix A
using the Bunch-Kaufman diagonal pivoting method. The partial
factorization has the form:
A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
( 0 U22 ) ( 0 D ) ( U12' U22' )
A = ( L11 0 ) ( D 0 ) ( L11' L21' ) if UPLO = 'L'
( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
DLASYF is an auxiliary routine called by DSYTRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
A22 (if UPLO = 'L').
Arguments
=========
UPLO (input) CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N (input) INTEGER
The order of the matrix A. N >= 0.
NB (input) INTEGER
The maximum number of columns of the matrix A that should be
factored. NB should be at least 2 to allow for 2-by-2 pivot
blocks.
KB (output) INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV (output) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U', only the last KB elements of IPIV are set;
if UPLO = 'L', only the first KB elements are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
W (workspace) DOUBLE PRECISION array, dimension (LDW,NB)
LDW (input) INTEGER
The leading dimension of the array W. LDW >= max(1,N).
INFO (output) INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular.
=====================================================================
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static doublereal c_b8 = -1.;
static doublereal c_b9 = 1.;
/* System generated locals */
integer a_dim1, a_offset, w_dim1, w_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer j, k;
static doublereal t, r1, d11, d21, d22;
static integer jb, jj, kk, jp, kp, kw, kkw, imax, jmax;
static doublereal alpha;
extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
integer *), dgemm_(char *, char *, integer *, integer *, integer *
, doublereal *, doublereal *, integer *, doublereal *, integer *,
doublereal *, doublereal *, integer *);
extern logical lsame_(char *, char *);
extern /* Subroutine */ int dgemv_(char *, integer *, integer *,
doublereal *, doublereal *, integer *, doublereal *, integer *,
doublereal *, doublereal *, integer *), dcopy_(integer *,
doublereal *, integer *, doublereal *, integer *), dswap_(integer
*, doublereal *, integer *, doublereal *, integer *);
static integer kstep;
static doublereal absakk;
extern integer idamax_(integer *, doublereal *, integer *);
static doublereal colmax, rowmax;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--ipiv;
w_dim1 = *ldw;
w_offset = 1 + w_dim1;
w -= w_offset;
/* Function Body */
*info = 0;
/* Initialize ALPHA for use in choosing pivot block size. */
alpha = (sqrt(17.) + 1.) / 8.;
if (lsame_(uplo, "U")) {
/* Factorize the trailing columns of A using the upper triangle
of A and working backwards, and compute the matrix W = U12*D
for use in updating A11
K is the main loop index, decreasing from N in steps of 1 or 2
KW is the column of W which corresponds to column K of A */
k = *n;
L10:
kw = *nb + k - *n;
/* Exit from loop */
if (k <= *n - *nb + 1 && *nb < *n || k < 1) {
goto L30;
}
/* Copy column K of A to column KW of W and update it */
dcopy_(&k, &a[k * a_dim1 + 1], &c__1, &w[kw * w_dim1 + 1], &c__1);
if (k < *n) {
i__1 = *n - k;
dgemv_("No transpose", &k, &i__1, &c_b8, &a[(k + 1) * a_dim1 + 1],
lda, &w[k + (kw + 1) * w_dim1], ldw, &c_b9, &w[kw *
w_dim1 + 1], &c__1);
}
kstep = 1;
/* Determine rows and columns to be interchanged and whether
a 1-by-1 or 2-by-2 pivot block will be used */
absakk = (d__1 = w[k + kw * w_dim1], abs(d__1));
/* IMAX is the row-index of the largest off-diagonal element in
column K, and COLMAX is its absolute value */
if (k > 1) {
i__1 = k - 1;
imax = idamax_(&i__1, &w[kw * w_dim1 + 1], &c__1);
colmax = (d__1 = w[imax + kw * w_dim1], abs(d__1));
} else {
colmax = 0.;
}
if (max(absakk,colmax) == 0.) {
/* Column K is zero: set INFO and continue */
if (*info == 0) {
*info = k;
}
kp = k;
} else {
if (absakk >= alpha * colmax) {
/* no interchange, use 1-by-1 pivot block */
kp = k;
} else {
/* Copy column IMAX to column KW-1 of W and update it */
dcopy_(&imax, &a[imax * a_dim1 + 1], &c__1, &w[(kw - 1) *
w_dim1 + 1], &c__1);
i__1 = k - imax;
dcopy_(&i__1, &a[imax + (imax + 1) * a_dim1], lda, &w[imax +
1 + (kw - 1) * w_dim1], &c__1);
if (k < *n) {
i__1 = *n - k;
dgemv_("No transpose", &k, &i__1, &c_b8, &a[(k + 1) *
a_dim1 + 1], lda, &w[imax + (kw + 1) * w_dim1],
ldw, &c_b9, &w[(kw - 1) * w_dim1 + 1], &c__1);
}
/* JMAX is the column-index of the largest off-diagonal
element in row IMAX, and ROWMAX is its absolute value */
i__1 = k - imax;
jmax = imax + idamax_(&i__1, &w[imax + 1 + (kw - 1) * w_dim1],
&c__1);
rowmax = (d__1 = w[jmax + (kw - 1) * w_dim1], abs(d__1));
if (imax > 1) {
i__1 = imax - 1;
jmax = idamax_(&i__1, &w[(kw - 1) * w_dim1 + 1], &c__1);
/* Computing MAX */
d__2 = rowmax, d__3 = (d__1 = w[jmax + (kw - 1) * w_dim1],
abs(d__1));
rowmax = max(d__2,d__3);
}
if (absakk >= alpha * colmax * (colmax / rowmax)) {
/* no interchange, use 1-by-1 pivot block */
kp = k;
} else if ((d__1 = w[imax + (kw - 1) * w_dim1], abs(d__1)) >=
alpha * rowmax) {
/* interchange rows and columns K and IMAX, use 1-by-1
pivot block */
kp = imax;
/* copy column KW-1 of W to column KW */
dcopy_(&k, &w[(kw - 1) * w_dim1 + 1], &c__1, &w[kw *
w_dim1 + 1], &c__1);
} else {
/* interchange rows and columns K-1 and IMAX, use 2-by-2
pivot block */
kp = imax;
kstep = 2;
}
}
kk = k - kstep + 1;
kkw = *nb + kk - *n;
/* Updated column KP is already stored in column KKW of W */
if (kp != kk) {
/* Copy non-updated column KK to column KP */
a[kp + k * a_dim1] = a[kk + k * a_dim1];
i__1 = k - 1 - kp;
dcopy_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + (kp +
1) * a_dim1], lda);
dcopy_(&kp, &a[kk * a_dim1 + 1], &c__1, &a[kp * a_dim1 + 1], &
c__1);
/* Interchange rows KK and KP in last KK columns of A and W */
i__1 = *n - kk + 1;
dswap_(&i__1, &a[kk + kk * a_dim1], lda, &a[kp + kk * a_dim1],
lda);
i__1 = *n - kk + 1;
dswap_(&i__1, &w[kk + kkw * w_dim1], ldw, &w[kp + kkw *
w_dim1], ldw);
}
if (kstep == 1) {
/* 1-by-1 pivot block D(k): column KW of W now holds
W(k) = U(k)*D(k)
where U(k) is the k-th column of U
Store U(k) in column k of A */
dcopy_(&k, &w[kw * w_dim1 + 1], &c__1, &a[k * a_dim1 + 1], &
c__1);
r1 = 1. / a[k + k * a_dim1];
i__1 = k - 1;
dscal_(&i__1, &r1, &a[k * a_dim1 + 1], &c__1);
} else {
/* 2-by-2 pivot block D(k): columns KW and KW-1 of W now
hold
( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
where U(k) and U(k-1) are the k-th and (k-1)-th columns
of U */
if (k > 2) {
/* Store U(k) and U(k-1) in columns k and k-1 of A */
d21 = w[k - 1 + kw * w_dim1];
d11 = w[k + kw * w_dim1] / d21;
d22 = w[k - 1 + (kw - 1) * w_dim1] / d21;
t = 1. / (d11 * d22 - 1.);
d21 = t / d21;
i__1 = k - 2;
for (j = 1; j <= i__1; ++j) {
a[j + (k - 1) * a_dim1] = d21 * (d11 * w[j + (kw - 1)
* w_dim1] - w[j + kw * w_dim1]);
a[j + k * a_dim1] = d21 * (d22 * w[j + kw * w_dim1] -
w[j + (kw - 1) * w_dim1]);
/* L20: */
}
}
/* Copy D(k) to A */
a[k - 1 + (k - 1) * a_dim1] = w[k - 1 + (kw - 1) * w_dim1];
a[k - 1 + k * a_dim1] = w[k - 1 + kw * w_dim1];
a[k + k * a_dim1] = w[k + kw * w_dim1];
}
}
/* Store details of the interchanges in IPIV */
if (kstep == 1) {
ipiv[k] = kp;
} else {
ipiv[k] = -kp;
ipiv[k - 1] = -kp;
}
/* Decrease K and return to the start of the main loop */
k -= kstep;
goto L10;
L30:
/* Update the upper triangle of A11 (= A(1:k,1:k)) as
A11 := A11 - U12*D*U12' = A11 - U12*W'
computing blocks of NB columns at a time */
i__1 = -(*nb);
for (j = (k - 1) / *nb * *nb + 1; i__1 < 0 ? j >= 1 : j <= 1; j +=
i__1) {
/* Computing MIN */
i__2 = *nb, i__3 = k - j + 1;
jb = min(i__2,i__3);
/* Update the upper triangle of the diagonal block */
i__2 = j + jb - 1;
for (jj = j; jj <= i__2; ++jj) {
i__3 = jj - j + 1;
i__4 = *n - k;
dgemv_("No transpose", &i__3, &i__4, &c_b8, &a[j + (k + 1) *
a_dim1], lda, &w[jj + (kw + 1) * w_dim1], ldw, &c_b9,
&a[j + jj * a_dim1], &c__1);
/* L40: */
}
/* Update the rectangular superdiagonal block */
i__2 = j - 1;
i__3 = *n - k;
dgemm_("No transpose", "Transpose", &i__2, &jb, &i__3, &c_b8, &a[(
k + 1) * a_dim1 + 1], lda, &w[j + (kw + 1) * w_dim1], ldw,
&c_b9, &a[j * a_dim1 + 1], lda);
/* L50: */
}
/* Put U12 in standard form by partially undoing the interchanges
in columns k+1:n */
j = k + 1;
L60:
jj = j;
jp = ipiv[j];
if (jp < 0) {
jp = -jp;
++j;
}
++j;
if (jp != jj && j <= *n) {
i__1 = *n - j + 1;
dswap_(&i__1, &a[jp + j * a_dim1], lda, &a[jj + j * a_dim1], lda);
}
if (j <= *n) {
goto L60;
}
/* Set KB to the number of columns factorized */
*kb = *n - k;
} else {
/* Factorize the leading columns of A using the lower triangle
of A and working forwards, and compute the matrix W = L21*D
for use in updating A22
K is the main loop index, increasing from 1 in steps of 1 or 2 */
k = 1;
L70:
/* Exit from loop */
if (k >= *nb && *nb < *n || k > *n) {
goto L90;
}
/* Copy column K of A to column K of W and update it */
i__1 = *n - k + 1;
dcopy_(&i__1, &a[k + k * a_dim1], &c__1, &w[k + k * w_dim1], &c__1);
i__1 = *n - k + 1;
i__2 = k - 1;
dgemv_("No transpose", &i__1, &i__2, &c_b8, &a[k + a_dim1], lda, &w[k
+ w_dim1], ldw, &c_b9, &w[k + k * w_dim1], &c__1);
kstep = 1;
/* Determine rows and columns to be interchanged and whether
a 1-by-1 or 2-by-2 pivot block will be used */
absakk = (d__1 = w[k + k * w_dim1], abs(d__1));
/* IMAX is the row-index of the largest off-diagonal element in
column K, and COLMAX is its absolute value */
if (k < *n) {
i__1 = *n - k;
imax = k + idamax_(&i__1, &w[k + 1 + k * w_dim1], &c__1);
colmax = (d__1 = w[imax + k * w_dim1], abs(d__1));
} else {
colmax = 0.;
}
if (max(absakk,colmax) == 0.) {
/* Column K is zero: set INFO and continue */
if (*info == 0) {
*info = k;
}
kp = k;
} else {
if (absakk >= alpha * colmax) {
/* no interchange, use 1-by-1 pivot block */
kp = k;
} else {
/* Copy column IMAX to column K+1 of W and update it */
i__1 = imax - k;
dcopy_(&i__1, &a[imax + k * a_dim1], lda, &w[k + (k + 1) *
w_dim1], &c__1);
i__1 = *n - imax + 1;
dcopy_(&i__1, &a[imax + imax * a_dim1], &c__1, &w[imax + (k +
1) * w_dim1], &c__1);
i__1 = *n - k + 1;
i__2 = k - 1;
dgemv_("No transpose", &i__1, &i__2, &c_b8, &a[k + a_dim1],
lda, &w[imax + w_dim1], ldw, &c_b9, &w[k + (k + 1) *
w_dim1], &c__1);
/* JMAX is the column-index of the largest off-diagonal
element in row IMAX, and ROWMAX is its absolute value */
i__1 = imax - k;
jmax = k - 1 + idamax_(&i__1, &w[k + (k + 1) * w_dim1], &c__1)
;
rowmax = (d__1 = w[jmax + (k + 1) * w_dim1], abs(d__1));
if (imax < *n) {
i__1 = *n - imax;
jmax = imax + idamax_(&i__1, &w[imax + 1 + (k + 1) *
w_dim1], &c__1);
/* Computing MAX */
d__2 = rowmax, d__3 = (d__1 = w[jmax + (k + 1) * w_dim1],
abs(d__1));
rowmax = max(d__2,d__3);
}
if (absakk >= alpha * colmax * (colmax / rowmax)) {
/* no interchange, use 1-by-1 pivot block */
kp = k;
} else if ((d__1 = w[imax + (k + 1) * w_dim1], abs(d__1)) >=
alpha * rowmax) {
/* interchange rows and columns K and IMAX, use 1-by-1
pivot block */
kp = imax;
/* copy column K+1 of W to column K */
i__1 = *n - k + 1;
dcopy_(&i__1, &w[k + (k + 1) * w_dim1], &c__1, &w[k + k *
w_dim1], &c__1);
} else {
/* interchange rows and columns K+1 and IMAX, use 2-by-2
pivot block */
kp = imax;
kstep = 2;
}
}
kk = k + kstep - 1;
/* Updated column KP is already stored in column KK of W */
if (kp != kk) {
/* Copy non-updated column KK to column KP */
a[kp + k * a_dim1] = a[kk + k * a_dim1];
i__1 = kp - k - 1;
dcopy_(&i__1, &a[k + 1 + kk * a_dim1], &c__1, &a[kp + (k + 1)
* a_dim1], lda);
i__1 = *n - kp + 1;
dcopy_(&i__1, &a[kp + kk * a_dim1], &c__1, &a[kp + kp *
a_dim1], &c__1);
/* Interchange rows KK and KP in first KK columns of A and W */
dswap_(&kk, &a[kk + a_dim1], lda, &a[kp + a_dim1], lda);
dswap_(&kk, &w[kk + w_dim1], ldw, &w[kp + w_dim1], ldw);
}
if (kstep == 1) {
/* 1-by-1 pivot block D(k): column k of W now holds
W(k) = L(k)*D(k)
where L(k) is the k-th column of L
Store L(k) in column k of A */
i__1 = *n - k + 1;
dcopy_(&i__1, &w[k + k * w_dim1], &c__1, &a[k + k * a_dim1], &
c__1);
if (k < *n) {
r1 = 1. / a[k + k * a_dim1];
i__1 = *n - k;
dscal_(&i__1, &r1, &a[k + 1 + k * a_dim1], &c__1);
}
} else {
/* 2-by-2 pivot block D(k): columns k and k+1 of W now hold
( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
where L(k) and L(k+1) are the k-th and (k+1)-th columns
of L */
if (k < *n - 1) {
/* Store L(k) and L(k+1) in columns k and k+1 of A */
d21 = w[k + 1 + k * w_dim1];
d11 = w[k + 1 + (k + 1) * w_dim1] / d21;
d22 = w[k + k * w_dim1] / d21;
t = 1. / (d11 * d22 - 1.);
d21 = t / d21;
i__1 = *n;
for (j = k + 2; j <= i__1; ++j) {
a[j + k * a_dim1] = d21 * (d11 * w[j + k * w_dim1] -
w[j + (k + 1) * w_dim1]);
a[j + (k + 1) * a_dim1] = d21 * (d22 * w[j + (k + 1) *
w_dim1] - w[j + k * w_dim1]);
/* L80: */
}
}
/* Copy D(k) to A */
a[k + k * a_dim1] = w[k + k * w_dim1];
a[k + 1 + k * a_dim1] = w[k + 1 + k * w_dim1];
a[k + 1 + (k + 1) * a_dim1] = w[k + 1 + (k + 1) * w_dim1];
}
}
/* Store details of the interchanges in IPIV */
if (kstep == 1) {
ipiv[k] = kp;
} else {
ipiv[k] = -kp;
ipiv[k + 1] = -kp;
}
/* Increase K and return to the start of the main loop */
k += kstep;
goto L70;
L90:
/* Update the lower triangle of A22 (= A(k:n,k:n)) as
A22 := A22 - L21*D*L21' = A22 - L21*W'
computing blocks of NB columns at a time */
i__1 = *n;
i__2 = *nb;
for (j = k; i__2 < 0 ? j >= i__1 : j <= i__1; j += i__2) {
/* Computing MIN */
i__3 = *nb, i__4 = *n - j + 1;
jb = min(i__3,i__4);
/* Update the lower triangle of the diagonal block */
i__3 = j + jb - 1;
for (jj = j; jj <= i__3; ++jj) {
i__4 = j + jb - jj;
i__5 = k - 1;
dgemv_("No transpose", &i__4, &i__5, &c_b8, &a[jj + a_dim1],
lda, &w[jj + w_dim1], ldw, &c_b9, &a[jj + jj * a_dim1]
, &c__1);
/* L100: */
}
/* Update the rectangular subdiagonal block */
if (j + jb <= *n) {
i__3 = *n - j - jb + 1;
i__4 = k - 1;
dgemm_("No transpose", "Transpose", &i__3, &jb, &i__4, &c_b8,
&a[j + jb + a_dim1], lda, &w[j + w_dim1], ldw, &c_b9,
&a[j + jb + j * a_dim1], lda);
}
/* L110: */
}
/* Put L21 in standard form by partially undoing the interchanges
in columns 1:k-1 */
j = k - 1;
L120:
jj = j;
jp = ipiv[j];
if (jp < 0) {
jp = -jp;
--j;
}
--j;
if (jp != jj && j >= 1) {
dswap_(&j, &a[jp + a_dim1], lda, &a[jj + a_dim1], lda);
}
if (j >= 1) {
goto L120;
}
/* Set KB to the number of columns factorized */
*kb = k - 1;
}
return 0;
/* End of DLASYF */
} /* dlasyf_ */