SUBROUTINE SSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, $ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, $ IFAIL, INFO ) * * -- LAPACK driver routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, ITYPE, IU, LDZ, M, N REAL ABSTOL, VL, VU * .. * .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) REAL AP( * ), BP( * ), W( * ), WORK( * ), $ Z( LDZ, * ) * .. * * Purpose * ======= * * SSPGVX computes selected eigenvalues, and optionally, 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 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) 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. * * VL (input) REAL * VU (input) REAL * 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) REAL * 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*SLAMCH('S'), not zero. * If this routine returns with INFO>0, indicating that some * eigenvectors did not converge, try setting ABSTOL to * 2*SLAMCH('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) REAL array, dimension (N) * On normal exit, the first M elements contain the selected * eigenvalues in ascending order. * * Z (output) REAL 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) REAL 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: SPPTRF or SSPEVX returned an error code: * <= N: if INFO = i, SSPEVX 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 .. LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ CHARACTER TRANS INTEGER J * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL SPPTRF, SSPEVX, SSPGST, STPMV, STPSV, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MIN * .. * .. Executable Statements .. * * Test the input parameters. * UPPER = LSAME( UPLO, 'U' ) WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) * INFO = 0 IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -3 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( VALEIG ) THEN IF( N.GT.0 .AND. VU.LE.VL ) THEN INFO = -9 END IF ELSE IF( INDEIG ) THEN IF( IL.LT.1 ) THEN INFO = -10 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -11 END IF END IF END IF IF( INFO.EQ.0 ) THEN IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -16 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SSPGVX', -INFO ) RETURN END IF * * Quick return if possible * M = 0 IF( N.EQ.0 ) $ RETURN * * Form a Cholesky factorization of B. * CALL SPPTRF( UPLO, N, BP, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF * * Transform problem to standard eigenvalue problem and solve. * CALL SSPGST( ITYPE, UPLO, N, AP, BP, INFO ) CALL SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, $ W, Z, LDZ, WORK, IWORK, IFAIL, INFO ) * IF( WANTZ ) THEN * * Backtransform eigenvectors to the original problem. * IF( INFO.GT.0 ) $ M = INFO - 1 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN * * For A*x=(lambda)*B*x and A*B*x=(lambda)*x; * backtransform eigenvectors: x = inv(L)'*y or inv(U)*y * IF( UPPER ) THEN TRANS = 'N' ELSE TRANS = 'T' END IF * DO 10 J = 1, M CALL STPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), $ 1 ) 10 CONTINUE * ELSE IF( ITYPE.EQ.3 ) THEN * * For B*A*x=(lambda)*x; * backtransform eigenvectors: x = L*y or U'*y * IF( UPPER ) THEN TRANS = 'T' ELSE TRANS = 'N' END IF * DO 20 J = 1, M CALL STPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), $ 1 ) 20 CONTINUE END IF END IF * RETURN * * End of SSPGVX * END