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opencv/3rdparty/lapack/dsytf2.c

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#include "clapack.h"
/* Subroutine */ int dsytf2_(char *uplo, integer *n, doublereal *a, integer *
lda, integer *ipiv, integer *info)
{
/* -- LAPACK routine (version 3.1) --
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
Purpose
=======
DSYTF2 computes the factorization of a real symmetric matrix A using
the Bunch-Kaufman diagonal pivoting method:
A = U*D*U' or A = L*D*L'
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, U' is the transpose of U, and D is symmetric and
block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
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.
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, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).
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 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.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.
Further Details
===============
09-29-06 - patch from
Bobby Cheng, MathWorks
Replace l.204 and l.372
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
by
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
1-96 - Based on modifications by J. Lewis, Boeing Computer Services
Company
If UPLO = 'U', then A = U*D*U', where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L', where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer i__, j, k;
static doublereal t, r1, d11, d12, d21, d22;
static integer kk, kp;
static doublereal wk, wkm1, wkp1;
static integer imax, jmax;
extern /* Subroutine */ int dsyr_(char *, integer *, doublereal *,
doublereal *, integer *, doublereal *, integer *);
static doublereal alpha;
extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
integer *);
extern logical lsame_(char *, char *);
extern /* Subroutine */ int dswap_(integer *, doublereal *, integer *,
doublereal *, integer *);
static integer kstep;
static logical upper;
static doublereal absakk;
extern integer idamax_(integer *, doublereal *, integer *);
extern logical disnan_(doublereal *);
extern /* Subroutine */ int xerbla_(char *, integer *);
static doublereal colmax, rowmax;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--ipiv;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSYTF2", &i__1);
return 0;
}
/* Initialize ALPHA for use in choosing pivot block size. */
alpha = (sqrt(17.) + 1.) / 8.;
if (upper) {
/* Factorize A as U*D*U' using the upper triangle of A
K is the main loop index, decreasing from N to 1 in steps of
1 or 2 */
k = *n;
L10:
/* If K < 1, exit from loop */
if (k < 1) {
goto L70;
}
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 = a[k + k * a_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, &a[k * a_dim1 + 1], &c__1);
colmax = (d__1 = a[imax + k * a_dim1], abs(d__1));
} else {
colmax = 0.;
}
if (max(absakk,colmax) == 0. || disnan_(&absakk)) {
/* Column K is zero or contains a NaN: 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 {
/* 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, &a[imax + (imax + 1) * a_dim1],
lda);
rowmax = (d__1 = a[imax + jmax * a_dim1], abs(d__1));
if (imax > 1) {
i__1 = imax - 1;
jmax = idamax_(&i__1, &a[imax * a_dim1 + 1], &c__1);
/* Computing MAX */
d__2 = rowmax, d__3 = (d__1 = a[jmax + imax * a_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 = a[imax + imax * a_dim1], abs(d__1)) >=
alpha * rowmax) {
/* interchange rows and columns K and IMAX, use 1-by-1
pivot block */
kp = imax;
} else {
/* interchange rows and columns K-1 and IMAX, use 2-by-2
pivot block */
kp = imax;
kstep = 2;
}
}
kk = k - kstep + 1;
if (kp != kk) {
/* Interchange rows and columns KK and KP in the leading
submatrix A(1:k,1:k) */
i__1 = kp - 1;
dswap_(&i__1, &a[kk * a_dim1 + 1], &c__1, &a[kp * a_dim1 + 1],
&c__1);
i__1 = kk - kp - 1;
dswap_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + (kp +
1) * a_dim1], lda);
t = a[kk + kk * a_dim1];
a[kk + kk * a_dim1] = a[kp + kp * a_dim1];
a[kp + kp * a_dim1] = t;
if (kstep == 2) {
t = a[k - 1 + k * a_dim1];
a[k - 1 + k * a_dim1] = a[kp + k * a_dim1];
a[kp + k * a_dim1] = t;
}
}
/* Update the leading submatrix */
if (kstep == 1) {
/* 1-by-1 pivot block D(k): column k now holds
W(k) = U(k)*D(k)
where U(k) is the k-th column of U
Perform a rank-1 update of A(1:k-1,1:k-1) as
A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)' */
r1 = 1. / a[k + k * a_dim1];
i__1 = k - 1;
d__1 = -r1;
dsyr_(uplo, &i__1, &d__1, &a[k * a_dim1 + 1], &c__1, &a[
a_offset], lda);
/* Store U(k) in column k */
i__1 = k - 1;
dscal_(&i__1, &r1, &a[k * a_dim1 + 1], &c__1);
} else {
/* 2-by-2 pivot block D(k): columns k and k-1 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
Perform a rank-2 update of A(1:k-2,1:k-2) as
A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )'
= A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )' */
if (k > 2) {
d12 = a[k - 1 + k * a_dim1];
d22 = a[k - 1 + (k - 1) * a_dim1] / d12;
d11 = a[k + k * a_dim1] / d12;
t = 1. / (d11 * d22 - 1.);
d12 = t / d12;
for (j = k - 2; j >= 1; --j) {
wkm1 = d12 * (d11 * a[j + (k - 1) * a_dim1] - a[j + k
* a_dim1]);
wk = d12 * (d22 * a[j + k * a_dim1] - a[j + (k - 1) *
a_dim1]);
for (i__ = j; i__ >= 1; --i__) {
a[i__ + j * a_dim1] = a[i__ + j * a_dim1] - a[i__
+ k * a_dim1] * wk - a[i__ + (k - 1) *
a_dim1] * wkm1;
/* L20: */
}
a[j + k * a_dim1] = wk;
a[j + (k - 1) * a_dim1] = wkm1;
/* L30: */
}
}
}
}
/* 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;
} else {
/* Factorize A as L*D*L' using the lower triangle of A
K is the main loop index, increasing from 1 to N in steps of
1 or 2 */
k = 1;
L40:
/* If K > N, exit from loop */
if (k > *n) {
goto L70;
}
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 = a[k + k * a_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, &a[k + 1 + k * a_dim1], &c__1);
colmax = (d__1 = a[imax + k * a_dim1], abs(d__1));
} else {
colmax = 0.;
}
if (max(absakk,colmax) == 0. || disnan_(&absakk)) {
/* Column K is zero or contains a NaN: 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 {
/* 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, &a[imax + k * a_dim1], lda);
rowmax = (d__1 = a[imax + jmax * a_dim1], abs(d__1));
if (imax < *n) {
i__1 = *n - imax;
jmax = imax + idamax_(&i__1, &a[imax + 1 + imax * a_dim1],
&c__1);
/* Computing MAX */
d__2 = rowmax, d__3 = (d__1 = a[jmax + imax * a_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 = a[imax + imax * a_dim1], abs(d__1)) >=
alpha * rowmax) {
/* interchange rows and columns K and IMAX, use 1-by-1
pivot block */
kp = imax;
} else {
/* interchange rows and columns K+1 and IMAX, use 2-by-2
pivot block */
kp = imax;
kstep = 2;
}
}
kk = k + kstep - 1;
if (kp != kk) {
/* Interchange rows and columns KK and KP in the trailing
submatrix A(k:n,k:n) */
if (kp < *n) {
i__1 = *n - kp;
dswap_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + 1
+ kp * a_dim1], &c__1);
}
i__1 = kp - kk - 1;
dswap_(&i__1, &a[kk + 1 + kk * a_dim1], &c__1, &a[kp + (kk +
1) * a_dim1], lda);
t = a[kk + kk * a_dim1];
a[kk + kk * a_dim1] = a[kp + kp * a_dim1];
a[kp + kp * a_dim1] = t;
if (kstep == 2) {
t = a[k + 1 + k * a_dim1];
a[k + 1 + k * a_dim1] = a[kp + k * a_dim1];
a[kp + k * a_dim1] = t;
}
}
/* Update the trailing submatrix */
if (kstep == 1) {
/* 1-by-1 pivot block D(k): column k now holds
W(k) = L(k)*D(k)
where L(k) is the k-th column of L */
if (k < *n) {
/* Perform a rank-1 update of A(k+1:n,k+1:n) as
A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)' */
d11 = 1. / a[k + k * a_dim1];
i__1 = *n - k;
d__1 = -d11;
dsyr_(uplo, &i__1, &d__1, &a[k + 1 + k * a_dim1], &c__1, &
a[k + 1 + (k + 1) * a_dim1], lda);
/* Store L(k) in column K */
i__1 = *n - k;
dscal_(&i__1, &d11, &a[k + 1 + k * a_dim1], &c__1);
}
} else {
/* 2-by-2 pivot block D(k) */
if (k < *n - 1) {
/* Perform a rank-2 update of A(k+2:n,k+2:n) as
A := A - ( (A(k) A(k+1))*D(k)**(-1) ) * (A(k) A(k+1))'
where L(k) and L(k+1) are the k-th and (k+1)-th
columns of L */
d21 = a[k + 1 + k * a_dim1];
d11 = a[k + 1 + (k + 1) * a_dim1] / d21;
d22 = a[k + k * a_dim1] / d21;
t = 1. / (d11 * d22 - 1.);
d21 = t / d21;
i__1 = *n;
for (j = k + 2; j <= i__1; ++j) {
wk = d21 * (d11 * a[j + k * a_dim1] - a[j + (k + 1) *
a_dim1]);
wkp1 = d21 * (d22 * a[j + (k + 1) * a_dim1] - a[j + k
* a_dim1]);
i__2 = *n;
for (i__ = j; i__ <= i__2; ++i__) {
a[i__ + j * a_dim1] = a[i__ + j * a_dim1] - a[i__
+ k * a_dim1] * wk - a[i__ + (k + 1) *
a_dim1] * wkp1;
/* L50: */
}
a[j + k * a_dim1] = wk;
a[j + (k + 1) * a_dim1] = wkp1;
/* L60: */
}
}
}
}
/* 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 L40;
}
L70:
return 0;
/* End of DSYTF2 */
} /* dsytf2_ */
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