/* Subroutine */ int sspgvd_(integer *itype, char *jobz, char *uplo, integer *
n, real *ap, real *bp, real *w, real *z__, integer *ldz, real *work,
integer *lwork, integer *iwork, integer *liwork, integer *info)
{
/* System generated locals */
integer z_dim1, z_offset, i__1;
real r__1, r__2;
/* Local variables */
integer j, neig;
extern logical lsame_(char *, char *);
integer lwmin;
char trans[1];
logical upper, wantz;
extern /* Subroutine */ int stpmv_(char *, char *, char *, integer *,
real *, real *, integer *), stpsv_(char *,
char *, char *, integer *, real *, real *, integer *), xerbla_(char *, integer *);
integer liwmin;
extern /* Subroutine */ int sspevd_(char *, char *, integer *, real *,
real *, real *, integer *, real *, integer *, integer *, integer *
, integer *), spptrf_(char *, integer *, real *,
integer *);
logical lquery;
extern /* Subroutine */ int sspgst_(integer *, char *, integer *, real *,
real *, integer *);
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SSPGVD computes all the eigenvalues, and optionally, the eigenvectors */
/* of a real generalized symmetric-definite eigenproblem, of the form */
/* A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and */
/* B are assumed to be symmetric, stored in packed format, and B is also */
/* positive definite. */
/* If eigenvectors are desired, it uses a divide and conquer algorithm. */
/* The divide and conquer algorithm makes very mild assumptions about */
/* floating point arithmetic. It will work on machines with a guard */
/* digit in add/subtract, or on those binary machines without guard */
/* digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or */
/* Cray-2. It could conceivably fail on hexadecimal or decimal machines */
/* without guard digits, but we know of none. */
/* Arguments */
/* ========= */
/* ITYPE (input) INTEGER */
/* Specifies the problem type to be solved: */
/* = 1: A*x = (lambda)*B*x */
/* = 2: A*B*x = (lambda)*x */
/* = 3: B*A*x = (lambda)*x */
/* JOBZ (input) CHARACTER*1 */
/* = 'N': Compute eigenvalues only; */
/* = 'V': Compute eigenvalues and eigenvectors. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangles of A and B are stored; */
/* = 'L': Lower triangles of A and B are stored. */
/* N (input) INTEGER */
/* The order of the matrices A and B. N >= 0. */
/* AP (input/output) REAL array, dimension (N*(N+1)/2) */
/* On entry, the upper or lower triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. */
/* On exit, the contents of AP are destroyed. */
/* BP (input/output) REAL array, dimension (N*(N+1)/2) */
/* On entry, the upper or lower triangle of the symmetric matrix */
/* B, packed columnwise in a linear array. The j-th column of B */
/* is stored in the array BP as follows: */
/* if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; */
/* if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. */
/* On exit, the triangular factor U or L from the Cholesky */
/* factorization B = U**T*U or B = L*L**T, in the same storage */
/* format as B. */
/* W (output) REAL array, dimension (N) */
/* If INFO = 0, the eigenvalues in ascending order. */
/* Z (output) REAL array, dimension (LDZ, N) */
/* If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of */
/* eigenvectors. The eigenvectors are normalized as follows: */
/* if ITYPE = 1 or 2, Z**T*B*Z = I; */
/* if ITYPE = 3, Z**T*inv(B)*Z = I. */
/* If JOBZ = 'N', then Z is not referenced. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1, and if */
/* JOBZ = 'V', LDZ >= max(1,N). */
/* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the required LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. */
/* If N <= 1, LWORK >= 1. */
/* If JOBZ = 'N' and N > 1, LWORK >= 2*N. */
/* If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the required sizes of the WORK and IWORK */
/* arrays, returns these values as the first entries of the WORK */
/* and IWORK arrays, and no error message related to LWORK or */
/* LIWORK is issued by XERBLA. */
/* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) */
/* On exit, if INFO = 0, IWORK(1) returns the required LIWORK. */
/* LIWORK (input) INTEGER */
/* The dimension of the array IWORK. */
/* If JOBZ = 'N' or N <= 1, LIWORK >= 1. */
/* If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. */
/* If LIWORK = -1, then a workspace query is assumed; the */
/* routine only calculates the required sizes of the WORK and */
/* IWORK arrays, returns these values as the first entries of */
/* the WORK and IWORK arrays, and no error message related to */
/* LWORK or LIWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: SPPTRF or SSPEVD returned an error code: */
/* <= N: if INFO = i, SSPEVD failed to converge; */
/* i off-diagonal elements of an intermediate */
/* tridiagonal form did not converge to zero; */
/* > N: if INFO = N + i, for 1 <= i <= N, then the leading */
/* minor of order i of B is not positive definite. */
/* The factorization of B could not be completed and */
/* no eigenvalues or eigenvectors were computed. */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--ap;
--bp;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
--iwork;
/* Function Body */
wantz = lsame_(jobz, "V");
upper = lsame_(uplo, "U");
lquery = *lwork == -1 || *liwork == -1;
*info = 0;
if (*itype < 1 || *itype > 3) {
*info = -1;
} else if (! (wantz || lsame_(jobz, "N"))) {
*info = -2;
} else if (! (upper || lsame_(uplo, "L"))) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*ldz < 1 || wantz && *ldz < *n) {
*info = -9;
}
if (*info == 0) {
if (*n <= 1) {
liwmin = 1;
lwmin = 1;
} else {
if (wantz) {
liwmin = *n * 5 + 3;
/* Computing 2nd power */
i__1 = *n;
lwmin = *n * 6 + 1 + (i__1 * i__1 << 1);
} else {
liwmin = 1;
lwmin = *n << 1;
}
}
work[1] = (real) lwmin;
iwork[1] = liwmin;
if (*lwork < lwmin && ! lquery) {
*info = -11;
} else if (*liwork < liwmin && ! lquery) {
*info = -13;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSPGVD", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Form a Cholesky factorization of BP. */
spptrf_(uplo, n, &bp[1], info);
if (*info != 0) {
*info = *n + *info;
return 0;
}
/* Transform problem to standard eigenvalue problem and solve. */
sspgst_(itype, uplo, n, &ap[1], &bp[1], info);
sspevd_(jobz, uplo, n, &ap[1], &w[1], &z__[z_offset], ldz, &work[1],
lwork, &iwork[1], liwork, info);
/* Computing MAX */
r__1 = (real) lwmin;
lwmin = dmax(r__1,work[1]);
/* Computing MAX */
r__1 = (real) liwmin, r__2 = (real) iwork[1];
liwmin = dmax(r__1,r__2);
if (wantz) {
/* Backtransform eigenvectors to the original problem. */
neig = *n;
if (*info > 0) {
neig = *info - 1;
}
if (*itype == 1 || *itype == 2) {
/* For A*x=(lambda)*B*x and A*B*x=(lambda)*x; */
/* backtransform eigenvectors: x = inv(L)'*y or inv(U)*y */
if (upper) {
*(unsigned char *)trans = 'N';
} else {
*(unsigned char *)trans = 'T';
}
i__1 = neig;
for (j = 1; j <= i__1; ++j) {
stpsv_(uplo, trans, "Non-unit", n, &bp[1], &z__[j * z_dim1 +
1], &c__1);
/* L10: */
}
} else if (*itype == 3) {
/* For B*A*x=(lambda)*x; */
/* backtransform eigenvectors: x = L*y or U'*y */
if (upper) {
*(unsigned char *)trans = 'T';
} else {
*(unsigned char *)trans = 'N';
}
i__1 = neig;
for (j = 1; j <= i__1; ++j) {
stpmv_(uplo, trans, "Non-unit", n, &bp[1], &z__[j * z_dim1 +
1], &c__1);
/* L20: */
}
}
}
work[1] = (real) lwmin;
iwork[1] = liwmin;
return 0;
/* End of SSPGVD */
} /* sspgvd_ */
/* Subroutine */ int sspgv_(integer *itype, char *jobz, char *uplo, integer *
n, real *ap, real *bp, real *w, real *z__, integer *ldz, real *work,
integer *info)
{
/* -- LAPACK driver 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
=======
SSPGV computes all the eigenvalues and, optionally, the eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x.
Here A and B are assumed to be symmetric, stored in packed format,
and B is also positive definite.
Arguments
=========
ITYPE (input) INTEGER
Specifies the problem type to be solved:
= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x
JOBZ (input) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
UPLO (input) CHARACTER*1
= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
AP (input/output) REAL array, dimension
(N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, the contents of AP are destroyed.
BP (input/output) REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
B, packed columnwise in a linear array. The j-th column of B
is stored in the array BP as follows:
if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
On exit, the triangular factor U or L from the Cholesky
factorization B = U**T*U or B = L*L**T, in the same storage
format as B.
W (output) REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) REAL array, dimension (LDZ, N)
If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
eigenvectors. The eigenvectors are normalized as follows:
if ITYPE = 1 or 2, Z**T*B*Z = I;
if ITYPE = 3, Z**T*inv(B)*Z = I.
If JOBZ = 'N', then Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = 'V', LDZ >= max(1,N).
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
> 0: SPPTRF or SSPEV returned an error code:
<= N: if INFO = i, SSPEV failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero.
> N: if INFO = n + i, for 1 <= i <= n, then the leading
minor of order i of B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* System generated locals */
integer z_dim1, z_offset, i__1;
/* Local variables */
static integer neig, j;
extern logical lsame_(char *, char *);
static char trans[1];
static logical upper;
extern /* Subroutine */ int sspev_(char *, char *, integer *, real *,
real *, real *, integer *, real *, integer *);
static logical wantz;
extern /* Subroutine */ int stpmv_(char *, char *, char *, integer *,
real *, real *, integer *), stpsv_(char *,
char *, char *, integer *, real *, real *, integer *), xerbla_(char *, integer *), spptrf_(char
*, integer *, real *, integer *), sspgst_(integer *, char
*, integer *, real *, real *, integer *);
#define z___ref(a_1,a_2) z__[(a_2)*z_dim1 + a_1]
--ap;
--bp;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--work;
/* Function Body */
wantz = lsame_(jobz, "V");
upper = lsame_(uplo, "U");
*info = 0;
if (*itype < 0 || *itype > 3) {
*info = -1;
} else if (! (wantz || lsame_(jobz, "N"))) {
*info = -2;
} else if (! (upper || lsame_(uplo, "L"))) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*ldz < 1 || wantz && *ldz < *n) {
*info = -9;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSPGV ", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Form a Cholesky factorization of B. */
spptrf_(uplo, n, &bp[1], info);
if (*info != 0) {
*info = *n + *info;
return 0;
}
/* Transform problem to standard eigenvalue problem and solve. */
sspgst_(itype, uplo, n, &ap[1], &bp[1], info);
sspev_(jobz, uplo, n, &ap[1], &w[1], &z__[z_offset], ldz, &work[1], info);
if (wantz) {
/* Backtransform eigenvectors to the original problem. */
neig = *n;
if (*info > 0) {
neig = *info - 1;
}
if (*itype == 1 || *itype == 2) {
/* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
backtransform eigenvectors: x = inv(L)'*y or inv(U)*y */
if (upper) {
*(unsigned char *)trans = 'N';
} else {
*(unsigned char *)trans = 'T';
}
i__1 = neig;
for (j = 1; j <= i__1; ++j) {
stpsv_(uplo, trans, "Non-unit", n, &bp[1], &z___ref(1, j), &
c__1);
/* L10: */
}
} else if (*itype == 3) {
/* For B*A*x=(lambda)*x;
backtransform eigenvectors: x = L*y or U'*y */
if (upper) {
*(unsigned char *)trans = 'T';
} else {
*(unsigned char *)trans = 'N';
}
i__1 = neig;
for (j = 1; j <= i__1; ++j) {
stpmv_(uplo, trans, "Non-unit", n, &bp[1], &z___ref(1, j), &
c__1);
/* L20: */
}
}
}
return 0;
/* End of SSPGV */
} /* sspgv_ */
int sspgv_(int *itype, char *jobz, char *uplo, int *
n, float *ap, float *bp, float *w, float *z__, int *ldz, float *work,
int *info)
{
/* System generated locals */
int z_dim1, z_offset, i__1;
/* Local variables */
int j, neig;
extern int lsame_(char *, char *);
char trans[1];
int upper;
extern int sspev_(char *, char *, int *, float *,
float *, float *, int *, float *, int *);
int wantz;
extern int stpmv_(char *, char *, char *, int *,
float *, float *, int *), stpsv_(char *,
char *, char *, int *, float *, float *, int *), xerbla_(char *, int *), spptrf_(char
*, int *, float *, int *), sspgst_(int *, char
*, int *, float *, float *, int *);
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SSPGV computes all the eigenvalues and, optionally, the eigenvectors */
/* of a float generalized symmetric-definite eigenproblem, of the form */
/* A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. */
/* Here A and B are assumed to be symmetric, stored in packed format, */
/* and B is also positive definite. */
/* Arguments */
/* ========= */
/* ITYPE (input) INTEGER */
/* Specifies the problem type to be solved: */
/* = 1: A*x = (lambda)*B*x */
/* = 2: A*B*x = (lambda)*x */
/* = 3: B*A*x = (lambda)*x */
/* JOBZ (input) CHARACTER*1 */
/* = 'N': Compute eigenvalues only; */
/* = 'V': Compute eigenvalues and eigenvectors. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangles of A and B are stored; */
/* = 'L': Lower triangles of A and B are stored. */
/* N (input) INTEGER */
/* The order of the matrices A and B. N >= 0. */
/* AP (input/output) REAL array, dimension */
/* (N*(N+1)/2) */
/* On entry, the upper or lower triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. */
/* On exit, the contents of AP are destroyed. */
/* BP (input/output) REAL array, dimension (N*(N+1)/2) */
/* On entry, the upper or lower triangle of the symmetric matrix */
/* B, packed columnwise in a linear array. The j-th column of B */
/* is stored in the array BP as follows: */
/* if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; */
/* if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. */
/* On exit, the triangular factor U or L from the Cholesky */
/* factorization B = U**T*U or B = L*L**T, in the same storage */
/* format as B. */
/* W (output) REAL array, dimension (N) */
/* If INFO = 0, the eigenvalues in ascending order. */
/* Z (output) REAL array, dimension (LDZ, N) */
/* If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of */
/* eigenvectors. The eigenvectors are normalized as follows: */
/* if ITYPE = 1 or 2, Z**T*B*Z = I; */
/* if ITYPE = 3, Z**T*inv(B)*Z = I. */
/* If JOBZ = 'N', then Z is not referenced. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1, and if */
/* JOBZ = 'V', LDZ >= MAX(1,N). */
/* 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 */
/* > 0: SPPTRF or SSPEV returned an error code: */
/* <= N: if INFO = i, SSPEV failed to converge; */
/* i off-diagonal elements of an intermediate */
/* tridiagonal form did not converge to zero. */
/* > N: if INFO = n + i, for 1 <= i <= n, then the leading */
/* minor of order i of B is not positive definite. */
/* The factorization of B could not be completed and */
/* no eigenvalues or eigenvectors were computed. */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--ap;
--bp;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
/* Function Body */
wantz = lsame_(jobz, "V");
upper = lsame_(uplo, "U");
*info = 0;
if (*itype < 1 || *itype > 3) {
*info = -1;
} else if (! (wantz || lsame_(jobz, "N"))) {
*info = -2;
} else if (! (upper || lsame_(uplo, "L"))) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*ldz < 1 || wantz && *ldz < *n) {
*info = -9;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSPGV ", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Form a Cholesky factorization of B. */
spptrf_(uplo, n, &bp[1], info);
if (*info != 0) {
*info = *n + *info;
return 0;
}
/* Transform problem to standard eigenvalue problem and solve. */
sspgst_(itype, uplo, n, &ap[1], &bp[1], info);
sspev_(jobz, uplo, n, &ap[1], &w[1], &z__[z_offset], ldz, &work[1], info);
if (wantz) {
/* Backtransform eigenvectors to the original problem. */
neig = *n;
if (*info > 0) {
neig = *info - 1;
}
if (*itype == 1 || *itype == 2) {
/* For A*x=(lambda)*B*x and A*B*x=(lambda)*x; */
/* backtransform eigenvectors: x = inv(L)'*y or inv(U)*y */
if (upper) {
*(unsigned char *)trans = 'N';
} else {
*(unsigned char *)trans = 'T';
}
i__1 = neig;
for (j = 1; j <= i__1; ++j) {
stpsv_(uplo, trans, "Non-unit", n, &bp[1], &z__[j * z_dim1 +
1], &c__1);
/* L10: */
}
} else if (*itype == 3) {
/* For B*A*x=(lambda)*x; */
/* backtransform eigenvectors: x = L*y or U'*y */
if (upper) {
*(unsigned char *)trans = 'T';
} else {
*(unsigned char *)trans = 'N';
}
i__1 = neig;
for (j = 1; j <= i__1; ++j) {
stpmv_(uplo, trans, "Non-unit", n, &bp[1], &z__[j * z_dim1 +
1], &c__1);
/* L20: */
}
}
}
return 0;
/* End of SSPGV */
} /* sspgv_ */
/* Subroutine */ int sspgvd_(integer *itype, char *jobz, char *uplo, integer *
n, real *ap, real *bp, real *w, real *z__, integer *ldz, real *work,
integer *lwork, integer *iwork, integer *liwork, integer *info)
{
/* -- LAPACK driver routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
Purpose
=======
SSPGVD computes all the eigenvalues, and optionally, the eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
B are assumed to be symmetric, stored in packed format, and B is also
positive definite.
If eigenvectors are desired, it uses a divide and conquer algorithm.
The divide and conquer algorithm makes very mild assumptions about
floating point arithmetic. It will work on machines with a guard
digit in add/subtract, or on those binary machines without guard
digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
Cray-2. It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Arguments
=========
ITYPE (input) INTEGER
Specifies the problem type to be solved:
= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x
JOBZ (input) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
UPLO (input) CHARACTER*1
= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
AP (input/output) REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, the contents of AP are destroyed.
BP (input/output) REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
B, packed columnwise in a linear array. The j-th column of B
is stored in the array BP as follows:
if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
On exit, the triangular factor U or L from the Cholesky
factorization B = U**T*U or B = L*L**T, in the same storage
format as B.
W (output) REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) REAL array, dimension (LDZ, N)
If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
eigenvectors. The eigenvectors are normalized as follows:
if ITYPE = 1 or 2, Z**T*B*Z = I;
if ITYPE = 3, Z**T*inv(B)*Z = I.
If JOBZ = 'N', then Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = 'V', LDZ >= max(1,N).
WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK.
If N <= 1, LWORK >= 1.
If JOBZ = 'N' and N > 1, LWORK >= 2*N.
If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK (workspace/output) INTEGER array, dimension (LIWORK)
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input) INTEGER
The dimension of the array IWORK.
If JOBZ = 'N' or N <= 1, LIWORK >= 1.
If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: SPPTRF or SSPEVD returned an error code:
<= N: if INFO = i, SSPEVD failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
> N: if INFO = N + i, for 1 <= i <= N, then the leading
minor of order i of B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
Further Details
===============
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__2 = 2;
static integer c__1 = 1;
/* System generated locals */
integer z_dim1, z_offset, i__1;
real r__1, r__2;
/* Builtin functions */
double log(doublereal);
integer pow_ii(integer *, integer *);
/* Local variables */
static integer neig, j;
extern logical lsame_(char *, char *);
static integer lwmin;
static char trans[1];
static logical upper, wantz;
extern /* Subroutine */ int stpmv_(char *, char *, char *, integer *,
real *, real *, integer *), stpsv_(char *,
char *, char *, integer *, real *, real *, integer *), xerbla_(char *, integer *);
static integer liwmin;
extern /* Subroutine */ int sspevd_(char *, char *, integer *, real *,
real *, real *, integer *, real *, integer *, integer *, integer *
, integer *), spptrf_(char *, integer *, real *,
integer *);
static logical lquery;
extern /* Subroutine */ int sspgst_(integer *, char *, integer *, real *,
real *, integer *);
static integer lgn;
#define z___ref(a_1,a_2) z__[(a_2)*z_dim1 + a_1]
--ap;
--bp;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--work;
--iwork;
/* Function Body */
wantz = lsame_(jobz, "V");
upper = lsame_(uplo, "U");
lquery = *lwork == -1 || *liwork == -1;
*info = 0;
if (*n <= 1) {
lgn = 0;
liwmin = 1;
lwmin = 1;
} else {
lgn = (integer) (log((real) (*n)) / log(2.f));
if (pow_ii(&c__2, &lgn) < *n) {
++lgn;
}
if (pow_ii(&c__2, &lgn) < *n) {
++lgn;
}
if (wantz) {
liwmin = *n * 5 + 3;
/* Computing 2nd power */
i__1 = *n;
lwmin = *n * 5 + 1 + (*n << 1) * lgn + (i__1 * i__1 << 1);
} else {
liwmin = 1;
lwmin = *n << 1;
}
}
if (*itype < 0 || *itype > 3) {
*info = -1;
} else if (! (wantz || lsame_(jobz, "N"))) {
*info = -2;
} else if (! (upper || lsame_(uplo, "L"))) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*ldz < max(1,*n)) {
*info = -9;
} else if (*lwork < lwmin && ! lquery) {
*info = -11;
} else if (*liwork < liwmin && ! lquery) {
*info = -13;
}
if (*info == 0) {
work[1] = (real) lwmin;
iwork[1] = liwmin;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSPGVD", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Form a Cholesky factorization of BP. */
spptrf_(uplo, n, &bp[1], info);
if (*info != 0) {
*info = *n + *info;
return 0;
}
/* Transform problem to standard eigenvalue problem and solve. */
sspgst_(itype, uplo, n, &ap[1], &bp[1], info);
sspevd_(jobz, uplo, n, &ap[1], &w[1], &z__[z_offset], ldz, &work[1],
lwork, &iwork[1], liwork, info);
/* Computing MAX */
r__1 = (real) lwmin;
lwmin = dmax(r__1,work[1]);
/* Computing MAX */
r__1 = (real) liwmin, r__2 = (real) iwork[1];
liwmin = dmax(r__1,r__2);
if (wantz) {
/* Backtransform eigenvectors to the original problem. */
neig = *n;
if (*info > 0) {
neig = *info - 1;
}
if (*itype == 1 || *itype == 2) {
/* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
backtransform eigenvectors: x = inv(L)'*y or inv(U)*y */
if (upper) {
*(unsigned char *)trans = 'N';
} else {
*(unsigned char *)trans = 'T';
}
i__1 = neig;
for (j = 1; j <= i__1; ++j) {
stpsv_(uplo, trans, "Non-unit", n, &bp[1], &z___ref(1, j), &
c__1);
/* L10: */
}
} else if (*itype == 3) {
/* For B*A*x=(lambda)*x;
backtransform eigenvectors: x = L*y or U'*y */
if (upper) {
*(unsigned char *)trans = 'T';
} else {
*(unsigned char *)trans = 'N';
}
i__1 = neig;
for (j = 1; j <= i__1; ++j) {
stpmv_(uplo, trans, "Non-unit", n, &bp[1], &z___ref(1, j), &
c__1);
/* L20: */
}
}
}
work[1] = (real) lwmin;
iwork[1] = liwmin;
return 0;
/* End of SSPGVD */
} /* sspgvd_ */