int dspgvx_(int *itype, char *jobz, char *range, char *
uplo, int *n, double *ap, double *bp, double *vl,
double *vu, int *il, int *iu, double *abstol, int
*m, double *w, double *z__, int *ldz, double *work,
int *iwork, int *ifail, int *info)
{
/* System generated locals */
int z_dim1, z_offset, i__1;
/* Local variables */
int j;
extern int lsame_(char *, char *);
char trans[1];
int upper;
extern int dtpmv_(char *, char *, char *, int *,
double *, double *, int *),
dtpsv_(char *, char *, char *, int *, double *,
double *, int *);
int wantz, alleig, indeig, valeig;
extern int xerbla_(char *, int *), dpptrf_(
char *, int *, double *, int *), dspgst_(
int *, char *, int *, double *, double *, int
*), dspevx_(char *, char *, char *, int *, double
*, double *, double *, int *, int *, double *,
int *, double *, double *, int *, double *,
int *, int *, int *);
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DSPGVX computes selected eigenvalues, and optionally, 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 storage, and B */
/* is also positive definite. Eigenvalues and eigenvectors can be */
/* selected by specifying either a range of values or a range of indices */
/* for the desired eigenvalues. */
/* 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. */
/* RANGE (input) CHARACTER*1 */
/* = 'A': all eigenvalues will be found. */
/* = 'V': all eigenvalues in the half-open interval (VL,VU] */
/* will be found. */
/* = 'I': the IL-th through IU-th eigenvalues will be found. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A and B are stored; */
/* = 'L': Lower triangle of A and B are stored. */
/* N (input) INTEGER */
/* The order of the matrix pencil (A,B). N >= 0. */
/* AP (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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. */
/* VL (input) DOUBLE PRECISION */
/* VU (input) DOUBLE PRECISION */
/* If RANGE='V', the lower and upper bounds of the interval to */
/* be searched for eigenvalues. VL < VU. */
/* Not referenced if RANGE = 'A' or 'I'. */
/* IL (input) INTEGER */
/* IU (input) INTEGER */
/* If RANGE='I', the indices (in ascending order) of the */
/* smallest and largest eigenvalues to be returned. */
/* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. */
/* Not referenced if RANGE = 'A' or 'V'. */
/* ABSTOL (input) DOUBLE PRECISION */
/* The absolute error tolerance for the eigenvalues. */
/* An approximate eigenvalue is accepted as converged */
/* when it is determined to lie in an interval [a,b] */
/* of width less than or equal to */
/* ABSTOL + EPS * MAX( |a|,|b| ) , */
/* where EPS is the machine precision. If ABSTOL is less than */
/* or equal to zero, then EPS*|T| will be used in its place, */
/* where |T| is the 1-norm of the tridiagonal matrix obtained */
/* by reducing A to tridiagonal form. */
/* Eigenvalues will be computed most accurately when ABSTOL is */
/* set to twice the underflow threshold 2*DLAMCH('S'), not zero. */
/* If this routine returns with INFO>0, indicating that some */
/* eigenvectors did not converge, try setting ABSTOL to */
/* 2*DLAMCH('S'). */
/* M (output) INTEGER */
/* The total number of eigenvalues found. 0 <= M <= N. */
/* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. */
/* W (output) DOUBLE PRECISION array, dimension (N) */
/* On normal exit, the first M elements contain the selected */
/* eigenvalues in ascending order. */
/* Z (output) DOUBLE PRECISION array, dimension (LDZ, MAX(1,M)) */
/* If JOBZ = 'N', then Z is not referenced. */
/* If JOBZ = 'V', then if INFO = 0, the first M columns of Z */
/* contain the orthonormal eigenvectors of the matrix A */
/* corresponding to the selected eigenvalues, with the i-th */
/* column of Z holding the eigenvector associated with W(i). */
/* 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 an eigenvector fails to converge, then that column of Z */
/* contains the latest approximation to the eigenvector, and the */
/* index of the eigenvector is returned in IFAIL. */
/* Note: the user must ensure that at least MAX(1,M) columns are */
/* supplied in the array Z; if RANGE = 'V', the exact value of M */
/* is not known in advance and an upper bound must be used. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1, and if */
/* JOBZ = 'V', LDZ >= MAX(1,N). */
/* WORK (workspace) DOUBLE PRECISION array, dimension (8*N) */
/* IWORK (workspace) INTEGER array, dimension (5*N) */
/* IFAIL (output) INTEGER array, dimension (N) */
/* If JOBZ = 'V', then if INFO = 0, the first M elements of */
/* IFAIL are zero. If INFO > 0, then IFAIL contains the */
/* indices of the eigenvectors that failed to converge. */
/* If JOBZ = 'N', then IFAIL is not referenced. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: DPPTRF or DSPEVX returned an error code: */
/* <= N: if INFO = i, DSPEVX failed to converge; */
/* i eigenvectors failed to converge. Their indices */
/* are stored in array IFAIL. */
/* > 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 */
/* ===================================================================== */
/* .. 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;
--ifail;
/* Function Body */
upper = lsame_(uplo, "U");
wantz = lsame_(jobz, "V");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
*info = 0;
if (*itype < 1 || *itype > 3) {
*info = -1;
} else if (! (wantz || lsame_(jobz, "N"))) {
*info = -2;
} else if (! (alleig || valeig || indeig)) {
*info = -3;
} else if (! (upper || lsame_(uplo, "L"))) {
*info = -4;
} else if (*n < 0) {
*info = -5;
} else {
if (valeig) {
if (*n > 0 && *vu <= *vl) {
*info = -9;
}
} else if (indeig) {
if (*il < 1) {
*info = -10;
} else if (*iu < MIN(*n,*il) || *iu > *n) {
*info = -11;
}
}
}
if (*info == 0) {
if (*ldz < 1 || wantz && *ldz < *n) {
*info = -16;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSPGVX", &i__1);
return 0;
}
/* Quick return if possible */
*m = 0;
if (*n == 0) {
return 0;
}
/* Form a Cholesky factorization of B. */
dpptrf_(uplo, n, &bp[1], info);
if (*info != 0) {
*info = *n + *info;
return 0;
}
/* Transform problem to standard eigenvalue problem and solve. */
dspgst_(itype, uplo, n, &ap[1], &bp[1], info);
dspevx_(jobz, range, uplo, n, &ap[1], vl, vu, il, iu, abstol, m, &w[1], &
z__[z_offset], ldz, &work[1], &iwork[1], &ifail[1], info);
if (wantz) {
/* Backtransform eigenvectors to the original problem. */
if (*info > 0) {
*m = *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 = *m;
for (j = 1; j <= i__1; ++j) {
dtpsv_(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 = *m;
for (j = 1; j <= i__1; ++j) {
dtpmv_(uplo, trans, "Non-unit", n, &bp[1], &z__[j * z_dim1 +
1], &c__1);
/* L20: */
}
}
}
return 0;
/* End of DSPGVX */
} /* dspgvx_ */
/* Subroutine */ int dspgvd_(integer *itype, char *jobz, char *uplo, integer *
n, doublereal *ap, doublereal *bp, doublereal *w, doublereal *z__,
integer *ldz, doublereal *work, integer *lwork, integer *iwork,
integer *liwork, integer *info)
{
/* System generated locals */
integer z_dim1, z_offset, i__1;
doublereal d__1, d__2;
/* Local variables */
integer j, neig;
extern logical lsame_(char *, char *);
integer lwmin;
char trans[1];
logical upper;
extern /* Subroutine */ int dtpmv_(char *, char *, char *, integer *,
doublereal *, doublereal *, integer *),
dtpsv_(char *, char *, char *, integer *, doublereal *,
doublereal *, integer *);
logical wantz;
extern /* Subroutine */ int xerbla_(char *, integer *), dspevd_(
char *, char *, integer *, doublereal *, doublereal *, doublereal
*, integer *, doublereal *, integer *, integer *, integer *,
integer *);
integer liwmin;
extern /* Subroutine */ int dpptrf_(char *, integer *, doublereal *,
integer *), dspgst_(integer *, char *, integer *,
doublereal *, doublereal *, integer *);
logical lquery;
/* -- LAPACK driver routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DSPGVD 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N) */
/* If INFO = 0, the eigenvalues in ascending order. */
/* Z (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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: DPPTRF or DSPEVD returned an error code: */
/* <= N: if INFO = i, DSPEVD 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] = (doublereal) lwmin;
iwork[1] = liwmin;
if (*lwork < lwmin && ! lquery) {
*info = -11;
} else if (*liwork < liwmin && ! lquery) {
*info = -13;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSPGVD", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Form a Cholesky factorization of BP. */
dpptrf_(uplo, n, &bp[1], info);
if (*info != 0) {
*info = *n + *info;
return 0;
}
/* Transform problem to standard eigenvalue problem and solve. */
dspgst_(itype, uplo, n, &ap[1], &bp[1], info);
dspevd_(jobz, uplo, n, &ap[1], &w[1], &z__[z_offset], ldz, &work[1],
lwork, &iwork[1], liwork, info);
/* Computing MAX */
d__1 = (doublereal) lwmin;
lwmin = (integer) max(d__1,work[1]);
/* Computing MAX */
d__1 = (doublereal) liwmin, d__2 = (doublereal) iwork[1];
liwmin = (integer) max(d__1,d__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) {
dtpsv_(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) {
dtpmv_(uplo, trans, "Non-unit", n, &bp[1], &z__[j * z_dim1 +
1], &c__1);
/* L20: */
}
}
}
work[1] = (doublereal) lwmin;
iwork[1] = liwmin;
return 0;
/* End of DSPGVD */
} /* dspgvd_ */
/* Subroutine */ int dspgv_(integer *itype, char *jobz, char *uplo, integer *
n, doublereal *ap, doublereal *bp, doublereal *w, doublereal *z__,
integer *ldz, doublereal *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
=======
DSPGV 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (3*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: DPPTRF or DSPEV returned an error code:
<= N: if INFO = i, DSPEV 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 *);
extern /* Subroutine */ int dspev_(char *, char *, integer *, doublereal *
, doublereal *, doublereal *, integer *, doublereal *, integer *);
static char trans[1];
static logical upper;
extern /* Subroutine */ int dtpmv_(char *, char *, char *, integer *,
doublereal *, doublereal *, integer *),
dtpsv_(char *, char *, char *, integer *, doublereal *,
doublereal *, integer *);
static logical wantz;
extern /* Subroutine */ int xerbla_(char *, integer *), dpptrf_(
char *, integer *, doublereal *, integer *), dspgst_(
integer *, char *, integer *, doublereal *, doublereal *, 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_("DSPGV ", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Form a Cholesky factorization of B. */
dpptrf_(uplo, n, &bp[1], info);
if (*info != 0) {
*info = *n + *info;
return 0;
}
/* Transform problem to standard eigenvalue problem and solve. */
dspgst_(itype, uplo, n, &ap[1], &bp[1], info);
dspev_(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) {
dtpsv_(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) {
dtpmv_(uplo, trans, "Non-unit", n, &bp[1], &z___ref(1, j), &
c__1);
/* L20: */
}
}
}
return 0;
/* End of DSPGV */
} /* dspgv_ */
/* Subroutine */ int dspgvd_(integer *itype, char *jobz, char *uplo, integer *
n, doublereal *ap, doublereal *bp, doublereal *w, doublereal *z__,
integer *ldz, doublereal *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
=======
DSPGVD 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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: DPPTRF or DSPEVD returned an error code:
<= N: if INFO = i, DSPEVD 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;
doublereal d__1, d__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;
extern /* Subroutine */ int dtpmv_(char *, char *, char *, integer *,
doublereal *, doublereal *, integer *),
dtpsv_(char *, char *, char *, integer *, doublereal *,
doublereal *, integer *);
static logical wantz;
extern /* Subroutine */ int xerbla_(char *, integer *), dspevd_(
char *, char *, integer *, doublereal *, doublereal *, doublereal
*, integer *, doublereal *, integer *, integer *, integer *,
integer *);
static integer liwmin;
extern /* Subroutine */ int dpptrf_(char *, integer *, doublereal *,
integer *), dspgst_(integer *, char *, integer *,
doublereal *, doublereal *, integer *);
static logical lquery;
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((doublereal) (*n)) / log(2.));
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] = (doublereal) lwmin;
iwork[1] = liwmin;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSPGVD", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Form a Cholesky factorization of BP. */
dpptrf_(uplo, n, &bp[1], info);
if (*info != 0) {
*info = *n + *info;
return 0;
}
/* Transform problem to standard eigenvalue problem and solve. */
dspgst_(itype, uplo, n, &ap[1], &bp[1], info);
dspevd_(jobz, uplo, n, &ap[1], &w[1], &z__[z_offset], ldz, &work[1],
lwork, &iwork[1], liwork, info);
/* Computing MAX */
d__1 = (doublereal) lwmin;
lwmin = (integer) max(d__1,work[1]);
/* Computing MAX */
d__1 = (doublereal) liwmin, d__2 = (doublereal) iwork[1];
liwmin = (integer) max(d__1,d__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) {
dtpsv_(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) {
dtpmv_(uplo, trans, "Non-unit", n, &bp[1], &z___ref(1, j), &
c__1);
/* L20: */
}
}
}
work[1] = (doublereal) lwmin;
iwork[1] = liwmin;
return 0;
/* End of DSPGVD */
} /* dspgvd_ */
