int
f2c_chpr(char* uplo, integer* N,
real* alpha,
complex* X, integer* incX,
complex* Ap)
{
chpr_(uplo, N, alpha,
X, incX, Ap);
return 0;
}
void
chpr(char uplo, int n, float alpha, complex *x, int incx, complex *ap )
{
chpr_( &uplo, &n, &alpha, x, &incx, ap );
}
/* Subroutine */ int chptrf_(char *uplo, integer *n, complex *ap, integer *
ipiv, integer *info, ftnlen uplo_len)
{
/* System generated locals */
integer i__1, i__2, i__3, i__4, i__5, i__6;
real r__1, r__2, r__3, r__4;
complex q__1, q__2, q__3, q__4, q__5, q__6;
/* Builtin functions */
double sqrt(doublereal), r_imag(complex *);
void r_cnjg(complex *, complex *);
/* Local variables */
static real d__;
static integer i__, j, k;
static complex t;
static real r1, d11;
static complex d12;
static real d22;
static complex d21;
static integer kc, kk, kp;
static complex wk;
static integer kx;
static real tt;
static integer knc, kpc, npp;
static complex wkm1, wkp1;
extern /* Subroutine */ int chpr_(char *, integer *, real *, complex *,
integer *, complex *, ftnlen);
static integer imax, jmax;
static real alpha;
extern logical lsame_(char *, char *, ftnlen, ftnlen);
extern /* Subroutine */ int cswap_(integer *, complex *, integer *,
complex *, integer *);
static integer kstep;
static logical upper;
extern doublereal slapy2_(real *, real *);
static real absakk;
extern integer icamax_(integer *, complex *, integer *);
extern /* Subroutine */ int csscal_(integer *, real *, complex *, integer
*), xerbla_(char *, integer *, ftnlen);
static real colmax, rowmax;
/* -- LAPACK routine (version 3.0) -- */
/* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
/* Courant Institute, Argonne National Lab, and Rice University */
/* June 30, 1999 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CHPTRF computes the factorization of a complex Hermitian packed */
/* matrix A using the Bunch-Kaufman diagonal pivoting method: */
/* A = U*D*U**H or A = L*D*L**H */
/* where U (or L) is a product of permutation and unit upper (lower) */
/* triangular matrices, and D is Hermitian and block diagonal with */
/* 1-by-1 and 2-by-2 diagonal blocks. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* AP (input/output) COMPLEX array, dimension (N*(N+1)/2) */
/* On entry, the upper or lower triangle of the Hermitian matrix */
/* A, packed columnwise in a linear array. The j-th column of A */
/* is stored in the array AP as follows: */
/* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */
/* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. */
/* On exit, the block diagonal matrix D and the multipliers used */
/* to obtain the factor U or L, stored as a packed triangular */
/* matrix overwriting A (see below for further details). */
/* 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 = -i, the i-th argument had an illegal value */
/* > 0: if INFO = i, D(i,i) 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 */
/* =============== */
/* 5-96 - 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). */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--ipiv;
--ap;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U", (ftnlen)1, (ftnlen)1);
if (! upper && ! lsame_(uplo, "L", (ftnlen)1, (ftnlen)1)) {
*info = -1;
} else if (*n < 0) {
*info = -2;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CHPTRF", &i__1, (ftnlen)6);
return 0;
}
/* Initialize ALPHA for use in choosing pivot block size. */
alpha = (sqrt(17.f) + 1.f) / 8.f;
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;
kc = (*n - 1) * *n / 2 + 1;
L10:
knc = kc;
/* If K < 1, exit from loop */
if (k < 1) {
goto L110;
}
kstep = 1;
/* Determine rows and columns to be interchanged and whether */
/* a 1-by-1 or 2-by-2 pivot block will be used */
i__1 = kc + k - 1;
absakk = (r__1 = ap[i__1].r, dabs(r__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 = icamax_(&i__1, &ap[kc], &c__1);
i__1 = kc + imax - 1;
colmax = (r__1 = ap[i__1].r, dabs(r__1)) + (r__2 = r_imag(&ap[kc
+ imax - 1]), dabs(r__2));
} else {
colmax = 0.f;
}
if (dmax(absakk,colmax) == 0.f) {
/* Column K is zero: set INFO and continue */
if (*info == 0) {
*info = k;
}
kp = k;
i__1 = kc + k - 1;
i__2 = kc + k - 1;
r__1 = ap[i__2].r;
ap[i__1].r = r__1, ap[i__1].i = 0.f;
} 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 */
rowmax = 0.f;
jmax = imax;
kx = imax * (imax + 1) / 2 + imax;
i__1 = k;
for (j = imax + 1; j <= i__1; ++j) {
i__2 = kx;
if ((r__1 = ap[i__2].r, dabs(r__1)) + (r__2 = r_imag(&ap[
kx]), dabs(r__2)) > rowmax) {
i__2 = kx;
rowmax = (r__1 = ap[i__2].r, dabs(r__1)) + (r__2 =
r_imag(&ap[kx]), dabs(r__2));
jmax = j;
}
kx += j;
/* L20: */
}
kpc = (imax - 1) * imax / 2 + 1;
if (imax > 1) {
i__1 = imax - 1;
jmax = icamax_(&i__1, &ap[kpc], &c__1);
/* Computing MAX */
i__1 = kpc + jmax - 1;
r__3 = rowmax, r__4 = (r__1 = ap[i__1].r, dabs(r__1)) + (
r__2 = r_imag(&ap[kpc + jmax - 1]), dabs(r__2));
rowmax = dmax(r__3,r__4);
}
if (absakk >= alpha * colmax * (colmax / rowmax)) {
/* no interchange, use 1-by-1 pivot block */
kp = k;
} else /* if(complicated condition) */ {
i__1 = kpc + imax - 1;
if ((r__1 = ap[i__1].r, dabs(r__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 (kstep == 2) {
knc = knc - k + 1;
}
if (kp != kk) {
/* Interchange rows and columns KK and KP in the leading */
/* submatrix A(1:k,1:k) */
i__1 = kp - 1;
cswap_(&i__1, &ap[knc], &c__1, &ap[kpc], &c__1);
kx = kpc + kp - 1;
i__1 = kk - 1;
for (j = kp + 1; j <= i__1; ++j) {
kx = kx + j - 1;
r_cnjg(&q__1, &ap[knc + j - 1]);
t.r = q__1.r, t.i = q__1.i;
i__2 = knc + j - 1;
r_cnjg(&q__1, &ap[kx]);
ap[i__2].r = q__1.r, ap[i__2].i = q__1.i;
i__2 = kx;
ap[i__2].r = t.r, ap[i__2].i = t.i;
/* L30: */
}
i__1 = kx + kk - 1;
r_cnjg(&q__1, &ap[kx + kk - 1]);
ap[i__1].r = q__1.r, ap[i__1].i = q__1.i;
i__1 = knc + kk - 1;
r1 = ap[i__1].r;
i__1 = knc + kk - 1;
i__2 = kpc + kp - 1;
r__1 = ap[i__2].r;
ap[i__1].r = r__1, ap[i__1].i = 0.f;
i__1 = kpc + kp - 1;
ap[i__1].r = r1, ap[i__1].i = 0.f;
if (kstep == 2) {
i__1 = kc + k - 1;
i__2 = kc + k - 1;
r__1 = ap[i__2].r;
ap[i__1].r = r__1, ap[i__1].i = 0.f;
i__1 = kc + k - 2;
t.r = ap[i__1].r, t.i = ap[i__1].i;
i__1 = kc + k - 2;
i__2 = kc + kp - 1;
ap[i__1].r = ap[i__2].r, ap[i__1].i = ap[i__2].i;
i__1 = kc + kp - 1;
ap[i__1].r = t.r, ap[i__1].i = t.i;
}
} else {
i__1 = kc + k - 1;
i__2 = kc + k - 1;
r__1 = ap[i__2].r;
ap[i__1].r = r__1, ap[i__1].i = 0.f;
if (kstep == 2) {
i__1 = kc - 1;
i__2 = kc - 1;
r__1 = ap[i__2].r;
ap[i__1].r = r__1, ap[i__1].i = 0.f;
}
}
/* 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)' */
i__1 = kc + k - 1;
r1 = 1.f / ap[i__1].r;
i__1 = k - 1;
r__1 = -r1;
chpr_(uplo, &i__1, &r__1, &ap[kc], &c__1, &ap[1], (ftnlen)1);
/* Store U(k) in column k */
i__1 = k - 1;
csscal_(&i__1, &r1, &ap[kc], &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) {
i__1 = k - 1 + (k - 1) * k / 2;
r__1 = ap[i__1].r;
r__2 = r_imag(&ap[k - 1 + (k - 1) * k / 2]);
d__ = slapy2_(&r__1, &r__2);
i__1 = k - 1 + (k - 2) * (k - 1) / 2;
d22 = ap[i__1].r / d__;
i__1 = k + (k - 1) * k / 2;
d11 = ap[i__1].r / d__;
tt = 1.f / (d11 * d22 - 1.f);
i__1 = k - 1 + (k - 1) * k / 2;
q__1.r = ap[i__1].r / d__, q__1.i = ap[i__1].i / d__;
d12.r = q__1.r, d12.i = q__1.i;
d__ = tt / d__;
for (j = k - 2; j >= 1; --j) {
i__1 = j + (k - 2) * (k - 1) / 2;
q__3.r = d11 * ap[i__1].r, q__3.i = d11 * ap[i__1].i;
r_cnjg(&q__5, &d12);
i__2 = j + (k - 1) * k / 2;
q__4.r = q__5.r * ap[i__2].r - q__5.i * ap[i__2].i,
q__4.i = q__5.r * ap[i__2].i + q__5.i * ap[
i__2].r;
q__2.r = q__3.r - q__4.r, q__2.i = q__3.i - q__4.i;
q__1.r = d__ * q__2.r, q__1.i = d__ * q__2.i;
wkm1.r = q__1.r, wkm1.i = q__1.i;
i__1 = j + (k - 1) * k / 2;
q__3.r = d22 * ap[i__1].r, q__3.i = d22 * ap[i__1].i;
i__2 = j + (k - 2) * (k - 1) / 2;
q__4.r = d12.r * ap[i__2].r - d12.i * ap[i__2].i,
q__4.i = d12.r * ap[i__2].i + d12.i * ap[i__2]
.r;
q__2.r = q__3.r - q__4.r, q__2.i = q__3.i - q__4.i;
q__1.r = d__ * q__2.r, q__1.i = d__ * q__2.i;
wk.r = q__1.r, wk.i = q__1.i;
for (i__ = j; i__ >= 1; --i__) {
i__1 = i__ + (j - 1) * j / 2;
i__2 = i__ + (j - 1) * j / 2;
i__3 = i__ + (k - 1) * k / 2;
r_cnjg(&q__4, &wk);
q__3.r = ap[i__3].r * q__4.r - ap[i__3].i *
q__4.i, q__3.i = ap[i__3].r * q__4.i + ap[
i__3].i * q__4.r;
q__2.r = ap[i__2].r - q__3.r, q__2.i = ap[i__2].i
- q__3.i;
i__4 = i__ + (k - 2) * (k - 1) / 2;
r_cnjg(&q__6, &wkm1);
q__5.r = ap[i__4].r * q__6.r - ap[i__4].i *
q__6.i, q__5.i = ap[i__4].r * q__6.i + ap[
i__4].i * q__6.r;
q__1.r = q__2.r - q__5.r, q__1.i = q__2.i -
q__5.i;
ap[i__1].r = q__1.r, ap[i__1].i = q__1.i;
/* L40: */
}
i__1 = j + (k - 1) * k / 2;
ap[i__1].r = wk.r, ap[i__1].i = wk.i;
i__1 = j + (k - 2) * (k - 1) / 2;
ap[i__1].r = wkm1.r, ap[i__1].i = wkm1.i;
i__1 = j + (j - 1) * j / 2;
i__2 = j + (j - 1) * j / 2;
r__1 = ap[i__2].r;
q__1.r = r__1, q__1.i = 0.f;
ap[i__1].r = q__1.r, ap[i__1].i = q__1.i;
/* L50: */
}
}
}
}
/* 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;
kc = knc - k;
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;
kc = 1;
npp = *n * (*n + 1) / 2;
L60:
knc = kc;
/* If K > N, exit from loop */
if (k > *n) {
goto L110;
}
kstep = 1;
/* Determine rows and columns to be interchanged and whether */
/* a 1-by-1 or 2-by-2 pivot block will be used */
i__1 = kc;
absakk = (r__1 = ap[i__1].r, dabs(r__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 + icamax_(&i__1, &ap[kc + 1], &c__1);
i__1 = kc + imax - k;
colmax = (r__1 = ap[i__1].r, dabs(r__1)) + (r__2 = r_imag(&ap[kc
+ imax - k]), dabs(r__2));
} else {
colmax = 0.f;
}
if (dmax(absakk,colmax) == 0.f) {
/* Column K is zero: set INFO and continue */
if (*info == 0) {
*info = k;
}
kp = k;
i__1 = kc;
i__2 = kc;
r__1 = ap[i__2].r;
ap[i__1].r = r__1, ap[i__1].i = 0.f;
} 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 */
rowmax = 0.f;
kx = kc + imax - k;
i__1 = imax - 1;
for (j = k; j <= i__1; ++j) {
i__2 = kx;
if ((r__1 = ap[i__2].r, dabs(r__1)) + (r__2 = r_imag(&ap[
kx]), dabs(r__2)) > rowmax) {
i__2 = kx;
rowmax = (r__1 = ap[i__2].r, dabs(r__1)) + (r__2 =
r_imag(&ap[kx]), dabs(r__2));
jmax = j;
}
kx = kx + *n - j;
/* L70: */
}
kpc = npp - (*n - imax + 1) * (*n - imax + 2) / 2 + 1;
if (imax < *n) {
i__1 = *n - imax;
jmax = imax + icamax_(&i__1, &ap[kpc + 1], &c__1);
/* Computing MAX */
i__1 = kpc + jmax - imax;
r__3 = rowmax, r__4 = (r__1 = ap[i__1].r, dabs(r__1)) + (
r__2 = r_imag(&ap[kpc + jmax - imax]), dabs(r__2))
;
rowmax = dmax(r__3,r__4);
}
if (absakk >= alpha * colmax * (colmax / rowmax)) {
/* no interchange, use 1-by-1 pivot block */
kp = k;
} else /* if(complicated condition) */ {
i__1 = kpc;
if ((r__1 = ap[i__1].r, dabs(r__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 (kstep == 2) {
knc = knc + *n - k + 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;
cswap_(&i__1, &ap[knc + kp - kk + 1], &c__1, &ap[kpc + 1],
&c__1);
}
kx = knc + kp - kk;
i__1 = kp - 1;
for (j = kk + 1; j <= i__1; ++j) {
kx = kx + *n - j + 1;
r_cnjg(&q__1, &ap[knc + j - kk]);
t.r = q__1.r, t.i = q__1.i;
i__2 = knc + j - kk;
r_cnjg(&q__1, &ap[kx]);
ap[i__2].r = q__1.r, ap[i__2].i = q__1.i;
i__2 = kx;
ap[i__2].r = t.r, ap[i__2].i = t.i;
/* L80: */
}
i__1 = knc + kp - kk;
r_cnjg(&q__1, &ap[knc + kp - kk]);
ap[i__1].r = q__1.r, ap[i__1].i = q__1.i;
i__1 = knc;
r1 = ap[i__1].r;
i__1 = knc;
i__2 = kpc;
r__1 = ap[i__2].r;
ap[i__1].r = r__1, ap[i__1].i = 0.f;
i__1 = kpc;
ap[i__1].r = r1, ap[i__1].i = 0.f;
if (kstep == 2) {
i__1 = kc;
i__2 = kc;
r__1 = ap[i__2].r;
ap[i__1].r = r__1, ap[i__1].i = 0.f;
i__1 = kc + 1;
t.r = ap[i__1].r, t.i = ap[i__1].i;
i__1 = kc + 1;
i__2 = kc + kp - k;
ap[i__1].r = ap[i__2].r, ap[i__1].i = ap[i__2].i;
i__1 = kc + kp - k;
ap[i__1].r = t.r, ap[i__1].i = t.i;
}
} else {
i__1 = kc;
i__2 = kc;
r__1 = ap[i__2].r;
ap[i__1].r = r__1, ap[i__1].i = 0.f;
if (kstep == 2) {
i__1 = knc;
i__2 = knc;
r__1 = ap[i__2].r;
ap[i__1].r = r__1, ap[i__1].i = 0.f;
}
}
/* 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)' */
i__1 = kc;
r1 = 1.f / ap[i__1].r;
i__1 = *n - k;
r__1 = -r1;
chpr_(uplo, &i__1, &r__1, &ap[kc + 1], &c__1, &ap[kc + *n
- k + 1], (ftnlen)1);
/* Store L(k) in column K */
i__1 = *n - k;
csscal_(&i__1, &r1, &ap[kc + 1], &c__1);
}
} else {
/* 2-by-2 pivot block D(k): columns K and K+1 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) {
/* Perform a rank-2 update of A(k+2:n,k+2:n) as */
/* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )' */
/* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )' */
/* where L(k) and L(k+1) are the k-th and (k+1)-th */
/* columns of L */
i__1 = k + 1 + (k - 1) * ((*n << 1) - k) / 2;
r__1 = ap[i__1].r;
r__2 = r_imag(&ap[k + 1 + (k - 1) * ((*n << 1) - k) / 2]);
d__ = slapy2_(&r__1, &r__2);
i__1 = k + 1 + k * ((*n << 1) - k - 1) / 2;
d11 = ap[i__1].r / d__;
i__1 = k + (k - 1) * ((*n << 1) - k) / 2;
d22 = ap[i__1].r / d__;
tt = 1.f / (d11 * d22 - 1.f);
i__1 = k + 1 + (k - 1) * ((*n << 1) - k) / 2;
q__1.r = ap[i__1].r / d__, q__1.i = ap[i__1].i / d__;
d21.r = q__1.r, d21.i = q__1.i;
d__ = tt / d__;
i__1 = *n;
for (j = k + 2; j <= i__1; ++j) {
i__2 = j + (k - 1) * ((*n << 1) - k) / 2;
q__3.r = d11 * ap[i__2].r, q__3.i = d11 * ap[i__2].i;
i__3 = j + k * ((*n << 1) - k - 1) / 2;
q__4.r = d21.r * ap[i__3].r - d21.i * ap[i__3].i,
q__4.i = d21.r * ap[i__3].i + d21.i * ap[i__3]
.r;
q__2.r = q__3.r - q__4.r, q__2.i = q__3.i - q__4.i;
q__1.r = d__ * q__2.r, q__1.i = d__ * q__2.i;
wk.r = q__1.r, wk.i = q__1.i;
i__2 = j + k * ((*n << 1) - k - 1) / 2;
q__3.r = d22 * ap[i__2].r, q__3.i = d22 * ap[i__2].i;
r_cnjg(&q__5, &d21);
i__3 = j + (k - 1) * ((*n << 1) - k) / 2;
q__4.r = q__5.r * ap[i__3].r - q__5.i * ap[i__3].i,
q__4.i = q__5.r * ap[i__3].i + q__5.i * ap[
i__3].r;
q__2.r = q__3.r - q__4.r, q__2.i = q__3.i - q__4.i;
q__1.r = d__ * q__2.r, q__1.i = d__ * q__2.i;
wkp1.r = q__1.r, wkp1.i = q__1.i;
i__2 = *n;
for (i__ = j; i__ <= i__2; ++i__) {
i__3 = i__ + (j - 1) * ((*n << 1) - j) / 2;
i__4 = i__ + (j - 1) * ((*n << 1) - j) / 2;
i__5 = i__ + (k - 1) * ((*n << 1) - k) / 2;
r_cnjg(&q__4, &wk);
q__3.r = ap[i__5].r * q__4.r - ap[i__5].i *
q__4.i, q__3.i = ap[i__5].r * q__4.i + ap[
i__5].i * q__4.r;
q__2.r = ap[i__4].r - q__3.r, q__2.i = ap[i__4].i
- q__3.i;
i__6 = i__ + k * ((*n << 1) - k - 1) / 2;
r_cnjg(&q__6, &wkp1);
q__5.r = ap[i__6].r * q__6.r - ap[i__6].i *
q__6.i, q__5.i = ap[i__6].r * q__6.i + ap[
i__6].i * q__6.r;
q__1.r = q__2.r - q__5.r, q__1.i = q__2.i -
q__5.i;
ap[i__3].r = q__1.r, ap[i__3].i = q__1.i;
/* L90: */
}
i__2 = j + (k - 1) * ((*n << 1) - k) / 2;
ap[i__2].r = wk.r, ap[i__2].i = wk.i;
i__2 = j + k * ((*n << 1) - k - 1) / 2;
ap[i__2].r = wkp1.r, ap[i__2].i = wkp1.i;
i__2 = j + (j - 1) * ((*n << 1) - j) / 2;
i__3 = j + (j - 1) * ((*n << 1) - j) / 2;
r__1 = ap[i__3].r;
q__1.r = r__1, q__1.i = 0.f;
ap[i__2].r = q__1.r, ap[i__2].i = q__1.i;
/* L100: */
}
}
}
}
/* 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;
kc = knc + *n - k + 2;
goto L60;
}
L110:
return 0;
/* End of CHPTRF */
} /* chptrf_ */
/* Subroutine */ int cpptrf_(char *uplo, integer *n, complex *ap, integer *
info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
CPPTRF computes the Cholesky factorization of a complex Hermitian
positive definite matrix A stored in packed format.
The factorization has the form
A = U**H * U, if UPLO = 'U', or
A = L * L**H, if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
AP (input/output) COMPLEX array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
See below for further details.
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**H*U or A = L*L**H, in the same
storage format as A.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not
positive definite, and the factorization could not be
completed.
Further Details
===============
The packed storage scheme is illustrated by the following example
when N = 4, UPLO = 'U':
Two-dimensional storage of the Hermitian matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = conjg(aji))
a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static real c_b16 = -1.f;
/* System generated locals */
integer i__1, i__2, i__3;
real r__1;
complex q__1, q__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
extern /* Subroutine */ int chpr_(char *, integer *, real *, complex *,
integer *, complex *);
static integer j;
extern /* Complex */ VOID cdotc_(complex *, integer *, complex *, integer
*, complex *, integer *);
extern logical lsame_(char *, char *);
static logical upper;
extern /* Subroutine */ int ctpsv_(char *, char *, char *, integer *,
complex *, complex *, integer *);
static integer jc, jj;
extern /* Subroutine */ int csscal_(integer *, real *, complex *, integer
*), xerbla_(char *, integer *);
static real ajj;
--ap;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CPPTRF", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
if (upper) {
/* Compute the Cholesky factorization A = U'*U. */
jj = 0;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
jc = jj + 1;
jj += j;
/* Compute elements 1:J-1 of column J. */
if (j > 1) {
i__2 = j - 1;
ctpsv_("Upper", "Conjugate transpose", "Non-unit", &i__2, &ap[
1], &ap[jc], &c__1);
}
/* Compute U(J,J) and test for non-positive-definiteness. */
i__2 = jj;
r__1 = ap[i__2].r;
i__3 = j - 1;
cdotc_(&q__2, &i__3, &ap[jc], &c__1, &ap[jc], &c__1);
q__1.r = r__1 - q__2.r, q__1.i = -q__2.i;
ajj = q__1.r;
if (ajj <= 0.f) {
i__2 = jj;
ap[i__2].r = ajj, ap[i__2].i = 0.f;
goto L30;
}
i__2 = jj;
r__1 = sqrt(ajj);
ap[i__2].r = r__1, ap[i__2].i = 0.f;
/* L10: */
}
} else {
/* Compute the Cholesky factorization A = L*L'. */
jj = 1;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Compute L(J,J) and test for non-positive-definiteness. */
i__2 = jj;
ajj = ap[i__2].r;
if (ajj <= 0.f) {
i__2 = jj;
ap[i__2].r = ajj, ap[i__2].i = 0.f;
goto L30;
}
ajj = sqrt(ajj);
i__2 = jj;
ap[i__2].r = ajj, ap[i__2].i = 0.f;
/* Compute elements J+1:N of column J and update the trailing
submatrix. */
if (j < *n) {
i__2 = *n - j;
r__1 = 1.f / ajj;
csscal_(&i__2, &r__1, &ap[jj + 1], &c__1);
i__2 = *n - j;
chpr_("Lower", &i__2, &c_b16, &ap[jj + 1], &c__1, &ap[jj + *n
- j + 1]);
jj = jj + *n - j + 1;
}
/* L20: */
}
}
goto L40;
L30:
*info = j;
L40:
return 0;
/* End of CPPTRF */
} /* cpptrf_ */
int cpptri_(char *uplo, int *n, complex *ap, int *
info)
{
/* System generated locals */
int i__1, i__2, i__3;
float r__1;
complex q__1;
/* Local variables */
int j, jc, jj;
float ajj;
int jjn;
extern int chpr_(char *, int *, float *, complex *,
int *, complex *);
extern /* Complex */ VOID cdotc_(complex *, int *, complex *, int
*, complex *, int *);
extern int lsame_(char *, char *);
extern int ctpmv_(char *, char *, char *, int *,
complex *, complex *, int *);
int upper;
extern int csscal_(int *, float *, complex *, int
*), xerbla_(char *, int *), ctptri_(char *, char *,
int *, complex *, int *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CPPTRI computes the inverse of a complex Hermitian positive definite */
/* matrix A using the Cholesky factorization A = U**H*U or A = L*L**H */
/* computed by CPPTRF. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangular factor is stored in AP; */
/* = 'L': Lower triangular factor is stored in AP. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* AP (input/output) COMPLEX array, dimension (N*(N+1)/2) */
/* On entry, the triangular factor U or L from the Cholesky */
/* factorization A = U**H*U or A = L*L**H, packed columnwise as */
/* a linear array. The j-th column of U or L is stored in the */
/* array AP as follows: */
/* if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j; */
/* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n. */
/* On exit, the upper or lower triangle of the (Hermitian) */
/* inverse of A, overwriting the input factor U or L. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = i, the (i,i) element of the factor U or L is */
/* zero, and the inverse could not be computed. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--ap;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CPPTRI", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Invert the triangular Cholesky factor U or L. */
ctptri_(uplo, "Non-unit", n, &ap[1], info);
if (*info > 0) {
return 0;
}
if (upper) {
/* Compute the product inv(U) * inv(U)'. */
jj = 0;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
jc = jj + 1;
jj += j;
if (j > 1) {
i__2 = j - 1;
chpr_("Upper", &i__2, &c_b8, &ap[jc], &c__1, &ap[1]);
}
i__2 = jj;
ajj = ap[i__2].r;
csscal_(&j, &ajj, &ap[jc], &c__1);
/* L10: */
}
} else {
/* Compute the product inv(L)' * inv(L). */
jj = 1;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
jjn = jj + *n - j + 1;
i__2 = jj;
i__3 = *n - j + 1;
cdotc_(&q__1, &i__3, &ap[jj], &c__1, &ap[jj], &c__1);
r__1 = q__1.r;
ap[i__2].r = r__1, ap[i__2].i = 0.f;
if (j < *n) {
i__2 = *n - j;
ctpmv_("Lower", "Conjugate transpose", "Non-unit", &i__2, &ap[
jjn], &ap[jj + 1], &c__1);
}
jj = jjn;
/* L20: */
}
}
return 0;
/* End of CPPTRI */
} /* cpptri_ */
/* Subroutine */ int cppt01_(char *uplo, integer *n, complex *a, complex *
afac, real *rwork, real *resid)
{
/* System generated locals */
integer i__1, i__2, i__3, i__4, i__5;
real r__1;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer i__, k, kc;
complex tc;
real tr, eps;
extern /* Subroutine */ int chpr_(char *, integer *, real *, complex *,
integer *, complex *), cscal_(integer *, complex *,
complex *, integer *);
extern /* Complex */ VOID cdotc_(complex *, integer *, complex *, integer
*, complex *, integer *);
extern logical lsame_(char *, char *);
real anorm;
extern /* Subroutine */ int ctpmv_(char *, char *, char *, integer *,
complex *, complex *, integer *);
extern doublereal clanhp_(char *, char *, integer *, complex *, real *), slamch_(char *);
/* -- LAPACK test routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CPPT01 reconstructs a Hermitian positive definite packed matrix A */
/* from its L*L' or U'*U factorization and computes the residual */
/* norm( L*L' - A ) / ( N * norm(A) * EPS ) or */
/* norm( U'*U - A ) / ( N * norm(A) * EPS ), */
/* where EPS is the machine epsilon, L' is the conjugate transpose of */
/* L, and U' is the conjugate transpose of U. */
/* Arguments */
/* ========== */
/* UPLO (input) CHARACTER*1 */
/* Specifies whether the upper or lower triangular part of the */
/* Hermitian matrix A is stored: */
/* = 'U': Upper triangular */
/* = 'L': Lower triangular */
/* N (input) INTEGER */
/* The number of rows and columns of the matrix A. N >= 0. */
/* A (input) COMPLEX array, dimension (N*(N+1)/2) */
/* The original Hermitian matrix A, stored as a packed */
/* triangular matrix. */
/* AFAC (input/output) COMPLEX array, dimension (N*(N+1)/2) */
/* On entry, the factor L or U from the L*L' or U'*U */
/* factorization of A, stored as a packed triangular matrix. */
/* Overwritten with the reconstructed matrix, and then with the */
/* difference L*L' - A (or U'*U - A). */
/* RWORK (workspace) REAL array, dimension (N) */
/* RESID (output) REAL */
/* If UPLO = 'L', norm(L*L' - A) / ( N * norm(A) * EPS ) */
/* If UPLO = 'U', norm(U'*U - A) / ( N * norm(A) * EPS ) */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Quick exit if N = 0 */
/* Parameter adjustments */
--rwork;
--afac;
--a;
/* Function Body */
if (*n <= 0) {
*resid = 0.f;
return 0;
}
/* Exit with RESID = 1/EPS if ANORM = 0. */
eps = slamch_("Epsilon");
anorm = clanhp_("1", uplo, n, &a[1], &rwork[1]);
if (anorm <= 0.f) {
*resid = 1.f / eps;
return 0;
}
/* Check the imaginary parts of the diagonal elements and return with */
/* an error code if any are nonzero. */
kc = 1;
if (lsame_(uplo, "U")) {
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (r_imag(&afac[kc]) != 0.f) {
*resid = 1.f / eps;
return 0;
}
kc = kc + k + 1;
/* L10: */
}
} else {
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (r_imag(&afac[kc]) != 0.f) {
*resid = 1.f / eps;
return 0;
}
kc = kc + *n - k + 1;
/* L20: */
}
}
/* Compute the product U'*U, overwriting U. */
if (lsame_(uplo, "U")) {
kc = *n * (*n - 1) / 2 + 1;
for (k = *n; k >= 1; --k) {
/* Compute the (K,K) element of the result. */
cdotc_(&q__1, &k, &afac[kc], &c__1, &afac[kc], &c__1);
tr = q__1.r;
i__1 = kc + k - 1;
afac[i__1].r = tr, afac[i__1].i = 0.f;
/* Compute the rest of column K. */
if (k > 1) {
i__1 = k - 1;
ctpmv_("Upper", "Conjugate", "Non-unit", &i__1, &afac[1], &
afac[kc], &c__1);
kc -= k - 1;
}
/* L30: */
}
/* Compute the difference L*L' - A */
kc = 1;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
i__2 = k - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = kc + i__ - 1;
i__4 = kc + i__ - 1;
i__5 = kc + i__ - 1;
q__1.r = afac[i__4].r - a[i__5].r, q__1.i = afac[i__4].i - a[
i__5].i;
afac[i__3].r = q__1.r, afac[i__3].i = q__1.i;
/* L40: */
}
i__2 = kc + k - 1;
i__3 = kc + k - 1;
i__4 = kc + k - 1;
r__1 = a[i__4].r;
q__1.r = afac[i__3].r - r__1, q__1.i = afac[i__3].i;
afac[i__2].r = q__1.r, afac[i__2].i = q__1.i;
kc += k;
/* L50: */
}
/* Compute the product L*L', overwriting L. */
} else {
kc = *n * (*n + 1) / 2;
for (k = *n; k >= 1; --k) {
/* Add a multiple of column K of the factor L to each of */
/* columns K+1 through N. */
if (k < *n) {
i__1 = *n - k;
chpr_("Lower", &i__1, &c_b19, &afac[kc + 1], &c__1, &afac[kc
+ *n - k + 1]);
}
/* Scale column K by the diagonal element. */
i__1 = kc;
tc.r = afac[i__1].r, tc.i = afac[i__1].i;
i__1 = *n - k + 1;
cscal_(&i__1, &tc, &afac[kc], &c__1);
kc -= *n - k + 2;
/* L60: */
}
/* Compute the difference U'*U - A */
kc = 1;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
i__2 = kc;
i__3 = kc;
i__4 = kc;
r__1 = a[i__4].r;
q__1.r = afac[i__3].r - r__1, q__1.i = afac[i__3].i;
afac[i__2].r = q__1.r, afac[i__2].i = q__1.i;
i__2 = *n;
for (i__ = k + 1; i__ <= i__2; ++i__) {
i__3 = kc + i__ - k;
i__4 = kc + i__ - k;
i__5 = kc + i__ - k;
q__1.r = afac[i__4].r - a[i__5].r, q__1.i = afac[i__4].i - a[
i__5].i;
afac[i__3].r = q__1.r, afac[i__3].i = q__1.i;
/* L70: */
}
kc = kc + *n - k + 1;
/* L80: */
}
}
/* Compute norm( L*U - A ) / ( N * norm(A) * EPS ) */
*resid = clanhp_("1", uplo, n, &afac[1], &rwork[1]);
*resid = *resid / (real) (*n) / anorm / eps;
return 0;
/* End of CPPT01 */
} /* cppt01_ */
