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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE SSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, 00002 $ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, 00003 $ IFAIL, INFO ) 00004 * 00005 * -- LAPACK driver routine (version 3.3.1) -- 00006 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00007 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00008 * -- April 2011 -- 00009 * 00010 * .. Scalar Arguments .. 00011 CHARACTER JOBZ, RANGE, UPLO 00012 INTEGER IL, INFO, ITYPE, IU, LDZ, M, N 00013 REAL ABSTOL, VL, VU 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IFAIL( * ), IWORK( * ) 00017 REAL AP( * ), BP( * ), W( * ), WORK( * ), 00018 $ Z( LDZ, * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * SSPGVX computes selected eigenvalues, and optionally, eigenvectors 00025 * of a real generalized symmetric-definite eigenproblem, of the form 00026 * A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A 00027 * and B are assumed to be symmetric, stored in packed storage, and B 00028 * is also positive definite. Eigenvalues and eigenvectors can be 00029 * selected by specifying either a range of values or a range of indices 00030 * for the desired eigenvalues. 00031 * 00032 * Arguments 00033 * ========= 00034 * 00035 * ITYPE (input) INTEGER 00036 * Specifies the problem type to be solved: 00037 * = 1: A*x = (lambda)*B*x 00038 * = 2: A*B*x = (lambda)*x 00039 * = 3: B*A*x = (lambda)*x 00040 * 00041 * JOBZ (input) CHARACTER*1 00042 * = 'N': Compute eigenvalues only; 00043 * = 'V': Compute eigenvalues and eigenvectors. 00044 * 00045 * RANGE (input) CHARACTER*1 00046 * = 'A': all eigenvalues will be found. 00047 * = 'V': all eigenvalues in the half-open interval (VL,VU] 00048 * will be found. 00049 * = 'I': the IL-th through IU-th eigenvalues will be found. 00050 * 00051 * UPLO (input) CHARACTER*1 00052 * = 'U': Upper triangle of A and B are stored; 00053 * = 'L': Lower triangle of A and B are stored. 00054 * 00055 * N (input) INTEGER 00056 * The order of the matrix pencil (A,B). N >= 0. 00057 * 00058 * AP (input/output) REAL array, dimension (N*(N+1)/2) 00059 * On entry, the upper or lower triangle of the symmetric matrix 00060 * A, packed columnwise in a linear array. The j-th column of A 00061 * is stored in the array AP as follows: 00062 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00063 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. 00064 * 00065 * On exit, the contents of AP are destroyed. 00066 * 00067 * BP (input/output) REAL array, dimension (N*(N+1)/2) 00068 * On entry, the upper or lower triangle of the symmetric matrix 00069 * B, packed columnwise in a linear array. The j-th column of B 00070 * is stored in the array BP as follows: 00071 * if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; 00072 * if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. 00073 * 00074 * On exit, the triangular factor U or L from the Cholesky 00075 * factorization B = U**T*U or B = L*L**T, in the same storage 00076 * format as B. 00077 * 00078 * VL (input) REAL 00079 * VU (input) REAL 00080 * If RANGE='V', the lower and upper bounds of the interval to 00081 * be searched for eigenvalues. VL < VU. 00082 * Not referenced if RANGE = 'A' or 'I'. 00083 * 00084 * IL (input) INTEGER 00085 * IU (input) INTEGER 00086 * If RANGE='I', the indices (in ascending order) of the 00087 * smallest and largest eigenvalues to be returned. 00088 * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 00089 * Not referenced if RANGE = 'A' or 'V'. 00090 * 00091 * ABSTOL (input) REAL 00092 * The absolute error tolerance for the eigenvalues. 00093 * An approximate eigenvalue is accepted as converged 00094 * when it is determined to lie in an interval [a,b] 00095 * of width less than or equal to 00096 * 00097 * ABSTOL + EPS * max( |a|,|b| ) , 00098 * 00099 * where EPS is the machine precision. If ABSTOL is less than 00100 * or equal to zero, then EPS*|T| will be used in its place, 00101 * where |T| is the 1-norm of the tridiagonal matrix obtained 00102 * by reducing A to tridiagonal form. 00103 * 00104 * Eigenvalues will be computed most accurately when ABSTOL is 00105 * set to twice the underflow threshold 2*SLAMCH('S'), not zero. 00106 * If this routine returns with INFO>0, indicating that some 00107 * eigenvectors did not converge, try setting ABSTOL to 00108 * 2*SLAMCH('S'). 00109 * 00110 * M (output) INTEGER 00111 * The total number of eigenvalues found. 0 <= M <= N. 00112 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 00113 * 00114 * W (output) REAL array, dimension (N) 00115 * On normal exit, the first M elements contain the selected 00116 * eigenvalues in ascending order. 00117 * 00118 * Z (output) REAL array, dimension (LDZ, max(1,M)) 00119 * If JOBZ = 'N', then Z is not referenced. 00120 * If JOBZ = 'V', then if INFO = 0, the first M columns of Z 00121 * contain the orthonormal eigenvectors of the matrix A 00122 * corresponding to the selected eigenvalues, with the i-th 00123 * column of Z holding the eigenvector associated with W(i). 00124 * The eigenvectors are normalized as follows: 00125 * if ITYPE = 1 or 2, Z**T*B*Z = I; 00126 * if ITYPE = 3, Z**T*inv(B)*Z = I. 00127 * 00128 * If an eigenvector fails to converge, then that column of Z 00129 * contains the latest approximation to the eigenvector, and the 00130 * index of the eigenvector is returned in IFAIL. 00131 * Note: the user must ensure that at least max(1,M) columns are 00132 * supplied in the array Z; if RANGE = 'V', the exact value of M 00133 * is not known in advance and an upper bound must be used. 00134 * 00135 * LDZ (input) INTEGER 00136 * The leading dimension of the array Z. LDZ >= 1, and if 00137 * JOBZ = 'V', LDZ >= max(1,N). 00138 * 00139 * WORK (workspace) REAL array, dimension (8*N) 00140 * 00141 * IWORK (workspace) INTEGER array, dimension (5*N) 00142 * 00143 * IFAIL (output) INTEGER array, dimension (N) 00144 * If JOBZ = 'V', then if INFO = 0, the first M elements of 00145 * IFAIL are zero. If INFO > 0, then IFAIL contains the 00146 * indices of the eigenvectors that failed to converge. 00147 * If JOBZ = 'N', then IFAIL is not referenced. 00148 * 00149 * INFO (output) INTEGER 00150 * = 0: successful exit 00151 * < 0: if INFO = -i, the i-th argument had an illegal value 00152 * > 0: SPPTRF or SSPEVX returned an error code: 00153 * <= N: if INFO = i, SSPEVX failed to converge; 00154 * i eigenvectors failed to converge. Their indices 00155 * are stored in array IFAIL. 00156 * > N: if INFO = N + i, for 1 <= i <= N, then the leading 00157 * minor of order i of B is not positive definite. 00158 * The factorization of B could not be completed and 00159 * no eigenvalues or eigenvectors were computed. 00160 * 00161 * Further Details 00162 * =============== 00163 * 00164 * Based on contributions by 00165 * Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA 00166 * 00167 * ===================================================================== 00168 * 00169 * .. Local Scalars .. 00170 LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ 00171 CHARACTER TRANS 00172 INTEGER J 00173 * .. 00174 * .. External Functions .. 00175 LOGICAL LSAME 00176 EXTERNAL LSAME 00177 * .. 00178 * .. External Subroutines .. 00179 EXTERNAL SPPTRF, SSPEVX, SSPGST, STPMV, STPSV, XERBLA 00180 * .. 00181 * .. Intrinsic Functions .. 00182 INTRINSIC MIN 00183 * .. 00184 * .. Executable Statements .. 00185 * 00186 * Test the input parameters. 00187 * 00188 UPPER = LSAME( UPLO, 'U' ) 00189 WANTZ = LSAME( JOBZ, 'V' ) 00190 ALLEIG = LSAME( RANGE, 'A' ) 00191 VALEIG = LSAME( RANGE, 'V' ) 00192 INDEIG = LSAME( RANGE, 'I' ) 00193 * 00194 INFO = 0 00195 IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN 00196 INFO = -1 00197 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 00198 INFO = -2 00199 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 00200 INFO = -3 00201 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN 00202 INFO = -4 00203 ELSE IF( N.LT.0 ) THEN 00204 INFO = -5 00205 ELSE 00206 IF( VALEIG ) THEN 00207 IF( N.GT.0 .AND. VU.LE.VL ) THEN 00208 INFO = -9 00209 END IF 00210 ELSE IF( INDEIG ) THEN 00211 IF( IL.LT.1 ) THEN 00212 INFO = -10 00213 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 00214 INFO = -11 00215 END IF 00216 END IF 00217 END IF 00218 IF( INFO.EQ.0 ) THEN 00219 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 00220 INFO = -16 00221 END IF 00222 END IF 00223 * 00224 IF( INFO.NE.0 ) THEN 00225 CALL XERBLA( 'SSPGVX', -INFO ) 00226 RETURN 00227 END IF 00228 * 00229 * Quick return if possible 00230 * 00231 M = 0 00232 IF( N.EQ.0 ) 00233 $ RETURN 00234 * 00235 * Form a Cholesky factorization of B. 00236 * 00237 CALL SPPTRF( UPLO, N, BP, INFO ) 00238 IF( INFO.NE.0 ) THEN 00239 INFO = N + INFO 00240 RETURN 00241 END IF 00242 * 00243 * Transform problem to standard eigenvalue problem and solve. 00244 * 00245 CALL SSPGST( ITYPE, UPLO, N, AP, BP, INFO ) 00246 CALL SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, 00247 $ W, Z, LDZ, WORK, IWORK, IFAIL, INFO ) 00248 * 00249 IF( WANTZ ) THEN 00250 * 00251 * Backtransform eigenvectors to the original problem. 00252 * 00253 IF( INFO.GT.0 ) 00254 $ M = INFO - 1 00255 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN 00256 * 00257 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x; 00258 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y 00259 * 00260 IF( UPPER ) THEN 00261 TRANS = 'N' 00262 ELSE 00263 TRANS = 'T' 00264 END IF 00265 * 00266 DO 10 J = 1, M 00267 CALL STPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), 00268 $ 1 ) 00269 10 CONTINUE 00270 * 00271 ELSE IF( ITYPE.EQ.3 ) THEN 00272 * 00273 * For B*A*x=(lambda)*x; 00274 * backtransform eigenvectors: x = L*y or U**T*y 00275 * 00276 IF( UPPER ) THEN 00277 TRANS = 'T' 00278 ELSE 00279 TRANS = 'N' 00280 END IF 00281 * 00282 DO 20 J = 1, M 00283 CALL STPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), 00284 $ 1 ) 00285 20 CONTINUE 00286 END IF 00287 END IF 00288 * 00289 RETURN 00290 * 00291 * End of SSPGVX 00292 * 00293 END
1.7.3 
