1 SUBROUTINE ZHPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, 2 $ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, 3 $ IWORK, IFAIL, INFO ) 4* 5* -- LAPACK driver routine (version 3.0) -- 6* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., 7* Courant Institute, Argonne National Lab, and Rice University 8* June 30, 1999 9* 10* .. Scalar Arguments .. 11 CHARACTER JOBZ, RANGE, UPLO 12 INTEGER IL, INFO, ITYPE, IU, LDZ, M, N 13 DOUBLE PRECISION ABSTOL, VL, VU 14* .. 15* .. Array Arguments .. 16 INTEGER IFAIL( * ), IWORK( * ) 17 DOUBLE PRECISION RWORK( * ), W( * ) 18 COMPLEX*16 AP( * ), BP( * ), WORK( * ), Z( LDZ, * ) 19* .. 20* 21* Purpose 22* ======= 23* 24* ZHPGVX computes selected eigenvalues and, optionally, eigenvectors 25* of a complex generalized Hermitian-definite eigenproblem, of the form 26* A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and 27* B are assumed to be Hermitian, stored in packed format, and B is also 28* positive definite. Eigenvalues and eigenvectors can be selected by 29* specifying either a range of values or a range of indices for the 30* desired eigenvalues. 31* 32* Arguments 33* ========= 34* 35* ITYPE (input) INTEGER 36* Specifies the problem type to be solved: 37* = 1: A*x = (lambda)*B*x 38* = 2: A*B*x = (lambda)*x 39* = 3: B*A*x = (lambda)*x 40* 41* JOBZ (input) CHARACTER*1 42* = 'N': Compute eigenvalues only; 43* = 'V': Compute eigenvalues and eigenvectors. 44* 45* RANGE (input) CHARACTER*1 46* = 'A': all eigenvalues will be found; 47* = 'V': all eigenvalues in the half-open interval (VL,VU] 48* will be found; 49* = 'I': the IL-th through IU-th eigenvalues will be found. 50* 51* UPLO (input) CHARACTER*1 52* = 'U': Upper triangles of A and B are stored; 53* = 'L': Lower triangles of A and B are stored. 54* 55* N (input) INTEGER 56* The order of the matrices A and B. N >= 0. 57* 58* AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2) 59* On entry, the upper or lower triangle of the Hermitian matrix 60* A, packed columnwise in a linear array. The j-th column of A 61* is stored in the array AP as follows: 62* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 63* if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. 64* 65* On exit, the contents of AP are destroyed. 66* 67* BP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2) 68* On entry, the upper or lower triangle of the Hermitian matrix 69* B, packed columnwise in a linear array. The j-th column of B 70* is stored in the array BP as follows: 71* if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; 72* if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. 73* 74* On exit, the triangular factor U or L from the Cholesky 75* factorization B = U**H*U or B = L*L**H, in the same storage 76* format as B. 77* 78* VL (input) DOUBLE PRECISION 79* VU (input) DOUBLE PRECISION 80* If RANGE='V', the lower and upper bounds of the interval to 81* be searched for eigenvalues. VL < VU. 82* Not referenced if RANGE = 'A' or 'I'. 83* 84* IL (input) INTEGER 85* IU (input) INTEGER 86* If RANGE='I', the indices (in ascending order) of the 87* smallest and largest eigenvalues to be returned. 88* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 89* Not referenced if RANGE = 'A' or 'V'. 90* 91* ABSTOL (input) DOUBLE PRECISION 92* The absolute error tolerance for the eigenvalues. 93* An approximate eigenvalue is accepted as converged 94* when it is determined to lie in an interval [a,b] 95* of width less than or equal to 96* 97* ABSTOL + EPS * max( |a|,|b| ) , 98* 99* where EPS is the machine precision. If ABSTOL is less than 100* or equal to zero, then EPS*|T| will be used in its place, 101* where |T| is the 1-norm of the tridiagonal matrix obtained 102* by reducing AP to tridiagonal form. 103* 104* Eigenvalues will be computed most accurately when ABSTOL is 105* set to twice the underflow threshold 2*DLAMCH('S'), not zero. 106* If this routine returns with INFO>0, indicating that some 107* eigenvectors did not converge, try setting ABSTOL to 108* 2*DLAMCH('S'). 109* 110* M (output) INTEGER 111* The total number of eigenvalues found. 0 <= M <= N. 112* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 113* 114* W (output) DOUBLE PRECISION array, dimension (N) 115* On normal exit, the first M elements contain the selected 116* eigenvalues in ascending order. 117* 118* Z (output) COMPLEX*16 array, dimension (LDZ, N) 119* If JOBZ = 'N', then Z is not referenced. 120* If JOBZ = 'V', then if INFO = 0, the first M columns of Z 121* contain the orthonormal eigenvectors of the matrix A 122* corresponding to the selected eigenvalues, with the i-th 123* column of Z holding the eigenvector associated with W(i). 124* The eigenvectors are normalized as follows: 125* if ITYPE = 1 or 2, Z**H*B*Z = I; 126* if ITYPE = 3, Z**H*inv(B)*Z = I. 127* 128* If an eigenvector fails to converge, then that column of Z 129* contains the latest approximation to the eigenvector, and the 130* index of the eigenvector is returned in IFAIL. 131* Note: the user must ensure that at least max(1,M) columns are 132* supplied in the array Z; if RANGE = 'V', the exact value of M 133* is not known in advance and an upper bound must be used. 134* 135* LDZ (input) INTEGER 136* The leading dimension of the array Z. LDZ >= 1, and if 137* JOBZ = 'V', LDZ >= max(1,N). 138* 139* WORK (workspace) COMPLEX*16 array, dimension (2*N) 140* 141* RWORK (workspace) DOUBLE PRECISION array, dimension (7*N) 142* 143* IWORK (workspace) INTEGER array, dimension (5*N) 144* 145* IFAIL (output) INTEGER array, dimension (N) 146* If JOBZ = 'V', then if INFO = 0, the first M elements of 147* IFAIL are zero. If INFO > 0, then IFAIL contains the 148* indices of the eigenvectors that failed to converge. 149* If JOBZ = 'N', then IFAIL is not referenced. 150* 151* INFO (output) INTEGER 152* = 0: successful exit 153* < 0: if INFO = -i, the i-th argument had an illegal value 154* > 0: ZPPTRF or ZHPEVX returned an error code: 155* <= N: if INFO = i, ZHPEVX failed to converge; 156* i eigenvectors failed to converge. Their indices 157* are stored in array IFAIL. 158* > N: if INFO = N + i, for 1 <= i <= n, then the leading 159* minor of order i of B is not positive definite. 160* The factorization of B could not be completed and 161* no eigenvalues or eigenvectors were computed. 162* 163* Further Details 164* =============== 165* 166* Based on contributions by 167* Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA 168* 169* ===================================================================== 170* 171* .. Local Scalars .. 172 LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ 173 CHARACTER TRANS 174 INTEGER J 175* .. 176* .. External Functions .. 177 LOGICAL LSAME 178 EXTERNAL LSAME 179* .. 180* .. External Subroutines .. 181 EXTERNAL XERBLA, ZHPEVX, ZHPGST, ZPPTRF, ZTPMV, ZTPSV 182* .. 183* .. Intrinsic Functions .. 184 INTRINSIC MIN 185* .. 186* .. Executable Statements .. 187* 188* Test the input parameters. 189* 190 WANTZ = LSAME( JOBZ, 'V' ) 191 UPPER = LSAME( UPLO, 'U' ) 192 ALLEIG = LSAME( RANGE, 'A' ) 193 VALEIG = LSAME( RANGE, 'V' ) 194 INDEIG = LSAME( RANGE, 'I' ) 195* 196 INFO = 0 197 IF( ITYPE.LT.0 .OR. ITYPE.GT.3 ) THEN 198 INFO = -1 199 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 200 INFO = -2 201 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 202 INFO = -3 203 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN 204 INFO = -4 205 ELSE IF( N.LT.0 ) THEN 206 INFO = -5 207 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN 208 INFO = -9 209 ELSE IF( INDEIG .AND. IL.LT.1 ) THEN 210 INFO = -10 211 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN 212 INFO = -11 213 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 214 INFO = -16 215 END IF 216 IF( INFO.NE.0 ) THEN 217 CALL XERBLA( 'ZHPGVX', -INFO ) 218 RETURN 219 END IF 220* 221* Quick return if possible 222* 223 IF( N.EQ.0 ) 224 $ RETURN 225* 226* Form a Cholesky factorization of B. 227* 228 CALL ZPPTRF( UPLO, N, BP, INFO ) 229 IF( INFO.NE.0 ) THEN 230 INFO = N + INFO 231 RETURN 232 END IF 233* 234* Transform problem to standard eigenvalue problem and solve. 235* 236 CALL ZHPGST( ITYPE, UPLO, N, AP, BP, INFO ) 237 CALL ZHPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, 238 $ W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO ) 239* 240 IF( WANTZ ) THEN 241* 242* Backtransform eigenvectors to the original problem. 243* 244 IF( INFO.GT.0 ) 245 $ M = INFO - 1 246 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN 247* 248* For A*x=(lambda)*B*x and A*B*x=(lambda)*x; 249* backtransform eigenvectors: x = inv(L)'*y or inv(U)*y 250* 251 IF( UPPER ) THEN 252 TRANS = 'N' 253 ELSE 254 TRANS = 'C' 255 END IF 256* 257 DO 10 J = 1, M 258 CALL ZTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), 259 $ 1 ) 260 10 CONTINUE 261* 262 ELSE IF( ITYPE.EQ.3 ) THEN 263* 264* For B*A*x=(lambda)*x; 265* backtransform eigenvectors: x = L*y or U'*y 266* 267 IF( UPPER ) THEN 268 TRANS = 'C' 269 ELSE 270 TRANS = 'N' 271 END IF 272* 273 DO 20 J = 1, M 274 CALL ZTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), 275 $ 1 ) 276 20 CONTINUE 277 END IF 278 END IF 279* 280 RETURN 281* 282* End of ZHPGVX 283* 284 END 285