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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, 2: $ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, 3: $ WORK, RWORK, INFO ) 4: * 5: * -- LAPACK driver routine (version 3.2) -- 6: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 8: * November 2006 9: * 10: * .. Scalar Arguments .. 11: CHARACTER EQUED, FACT, UPLO 12: INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS 13: DOUBLE PRECISION RCOND 14: * .. 15: * .. Array Arguments .. 16: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * ) 17: COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 18: $ WORK( * ), X( LDX, * ) 19: * .. 20: * 21: * Purpose 22: * ======= 23: * 24: * ZPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to 25: * compute the solution to a complex system of linear equations 26: * A * X = B, 27: * where A is an N-by-N Hermitian positive definite band matrix and X 28: * and B are N-by-NRHS matrices. 29: * 30: * Error bounds on the solution and a condition estimate are also 31: * provided. 32: * 33: * Description 34: * =========== 35: * 36: * The following steps are performed: 37: * 38: * 1. If FACT = 'E', real scaling factors are computed to equilibrate 39: * the system: 40: * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B 41: * Whether or not the system will be equilibrated depends on the 42: * scaling of the matrix A, but if equilibration is used, A is 43: * overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 44: * 45: * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to 46: * factor the matrix A (after equilibration if FACT = 'E') as 47: * A = U**H * U, if UPLO = 'U', or 48: * A = L * L**H, if UPLO = 'L', 49: * where U is an upper triangular band matrix, and L is a lower 50: * triangular band matrix. 51: * 52: * 3. If the leading i-by-i principal minor is not positive definite, 53: * then the routine returns with INFO = i. Otherwise, the factored 54: * form of A is used to estimate the condition number of the matrix 55: * A. If the reciprocal of the condition number is less than machine 56: * precision, INFO = N+1 is returned as a warning, but the routine 57: * still goes on to solve for X and compute error bounds as 58: * described below. 59: * 60: * 4. The system of equations is solved for X using the factored form 61: * of A. 62: * 63: * 5. Iterative refinement is applied to improve the computed solution 64: * matrix and calculate error bounds and backward error estimates 65: * for it. 66: * 67: * 6. If equilibration was used, the matrix X is premultiplied by 68: * diag(S) so that it solves the original system before 69: * equilibration. 70: * 71: * Arguments 72: * ========= 73: * 74: * FACT (input) CHARACTER*1 75: * Specifies whether or not the factored form of the matrix A is 76: * supplied on entry, and if not, whether the matrix A should be 77: * equilibrated before it is factored. 78: * = 'F': On entry, AFB contains the factored form of A. 79: * If EQUED = 'Y', the matrix A has been equilibrated 80: * with scaling factors given by S. AB and AFB will not 81: * be modified. 82: * = 'N': The matrix A will be copied to AFB and factored. 83: * = 'E': The matrix A will be equilibrated if necessary, then 84: * copied to AFB and factored. 85: * 86: * UPLO (input) CHARACTER*1 87: * = 'U': Upper triangle of A is stored; 88: * = 'L': Lower triangle of A is stored. 89: * 90: * N (input) INTEGER 91: * The number of linear equations, i.e., the order of the 92: * matrix A. N >= 0. 93: * 94: * KD (input) INTEGER 95: * The number of superdiagonals of the matrix A if UPLO = 'U', 96: * or the number of subdiagonals if UPLO = 'L'. KD >= 0. 97: * 98: * NRHS (input) INTEGER 99: * The number of right-hand sides, i.e., the number of columns 100: * of the matrices B and X. NRHS >= 0. 101: * 102: * AB (input/output) COMPLEX*16 array, dimension (LDAB,N) 103: * On entry, the upper or lower triangle of the Hermitian band 104: * matrix A, stored in the first KD+1 rows of the array, except 105: * if FACT = 'F' and EQUED = 'Y', then A must contain the 106: * equilibrated matrix diag(S)*A*diag(S). The j-th column of A 107: * is stored in the j-th column of the array AB as follows: 108: * if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j; 109: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD). 110: * See below for further details. 111: * 112: * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 113: * diag(S)*A*diag(S). 114: * 115: * LDAB (input) INTEGER 116: * The leading dimension of the array A. LDAB >= KD+1. 117: * 118: * AFB (input or output) COMPLEX*16 array, dimension (LDAFB,N) 119: * If FACT = 'F', then AFB is an input argument and on entry 120: * contains the triangular factor U or L from the Cholesky 121: * factorization A = U**H*U or A = L*L**H of the band matrix 122: * A, in the same storage format as A (see AB). If EQUED = 'Y', 123: * then AFB is the factored form of the equilibrated matrix A. 124: * 125: * If FACT = 'N', then AFB is an output argument and on exit 126: * returns the triangular factor U or L from the Cholesky 127: * factorization A = U**H*U or A = L*L**H. 128: * 129: * If FACT = 'E', then AFB is an output argument and on exit 130: * returns the triangular factor U or L from the Cholesky 131: * factorization A = U**H*U or A = L*L**H of the equilibrated 132: * matrix A (see the description of A for the form of the 133: * equilibrated matrix). 134: * 135: * LDAFB (input) INTEGER 136: * The leading dimension of the array AFB. LDAFB >= KD+1. 137: * 138: * EQUED (input or output) CHARACTER*1 139: * Specifies the form of equilibration that was done. 140: * = 'N': No equilibration (always true if FACT = 'N'). 141: * = 'Y': Equilibration was done, i.e., A has been replaced by 142: * diag(S) * A * diag(S). 143: * EQUED is an input argument if FACT = 'F'; otherwise, it is an 144: * output argument. 145: * 146: * S (input or output) DOUBLE PRECISION array, dimension (N) 147: * The scale factors for A; not accessed if EQUED = 'N'. S is 148: * an input argument if FACT = 'F'; otherwise, S is an output 149: * argument. If FACT = 'F' and EQUED = 'Y', each element of S 150: * must be positive. 151: * 152: * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS) 153: * On entry, the N-by-NRHS right hand side matrix B. 154: * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', 155: * B is overwritten by diag(S) * B. 156: * 157: * LDB (input) INTEGER 158: * The leading dimension of the array B. LDB >= max(1,N). 159: * 160: * X (output) COMPLEX*16 array, dimension (LDX,NRHS) 161: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to 162: * the original system of equations. Note that if EQUED = 'Y', 163: * A and B are modified on exit, and the solution to the 164: * equilibrated system is inv(diag(S))*X. 165: * 166: * LDX (input) INTEGER 167: * The leading dimension of the array X. LDX >= max(1,N). 168: * 169: * RCOND (output) DOUBLE PRECISION 170: * The estimate of the reciprocal condition number of the matrix 171: * A after equilibration (if done). If RCOND is less than the 172: * machine precision (in particular, if RCOND = 0), the matrix 173: * is singular to working precision. This condition is 174: * indicated by a return code of INFO > 0. 175: * 176: * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 177: * The estimated forward error bound for each solution vector 178: * X(j) (the j-th column of the solution matrix X). 179: * If XTRUE is the true solution corresponding to X(j), FERR(j) 180: * is an estimated upper bound for the magnitude of the largest 181: * element in (X(j) - XTRUE) divided by the magnitude of the 182: * largest element in X(j). The estimate is as reliable as 183: * the estimate for RCOND, and is almost always a slight 184: * overestimate of the true error. 185: * 186: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 187: * The componentwise relative backward error of each solution 188: * vector X(j) (i.e., the smallest relative change in 189: * any element of A or B that makes X(j) an exact solution). 190: * 191: * WORK (workspace) COMPLEX*16 array, dimension (2*N) 192: * 193: * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 194: * 195: * INFO (output) INTEGER 196: * = 0: successful exit 197: * < 0: if INFO = -i, the i-th argument had an illegal value 198: * > 0: if INFO = i, and i is 199: * <= N: the leading minor of order i of A is 200: * not positive definite, so the factorization 201: * could not be completed, and the solution has not 202: * been computed. RCOND = 0 is returned. 203: * = N+1: U is nonsingular, but RCOND is less than machine 204: * precision, meaning that the matrix is singular 205: * to working precision. Nevertheless, the 206: * solution and error bounds are computed because 207: * there are a number of situations where the 208: * computed solution can be more accurate than the 209: * value of RCOND would suggest. 210: * 211: * Further Details 212: * =============== 213: * 214: * The band storage scheme is illustrated by the following example, when 215: * N = 6, KD = 2, and UPLO = 'U': 216: * 217: * Two-dimensional storage of the Hermitian matrix A: 218: * 219: * a11 a12 a13 220: * a22 a23 a24 221: * a33 a34 a35 222: * a44 a45 a46 223: * a55 a56 224: * (aij=conjg(aji)) a66 225: * 226: * Band storage of the upper triangle of A: 227: * 228: * * * a13 a24 a35 a46 229: * * a12 a23 a34 a45 a56 230: * a11 a22 a33 a44 a55 a66 231: * 232: * Similarly, if UPLO = 'L' the format of A is as follows: 233: * 234: * a11 a22 a33 a44 a55 a66 235: * a21 a32 a43 a54 a65 * 236: * a31 a42 a53 a64 * * 237: * 238: * Array elements marked * are not used by the routine. 239: * 240: * ===================================================================== 241: * 242: * .. Parameters .. 243: DOUBLE PRECISION ZERO, ONE 244: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 245: * .. 246: * .. Local Scalars .. 247: LOGICAL EQUIL, NOFACT, RCEQU, UPPER 248: INTEGER I, INFEQU, J, J1, J2 249: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM 250: * .. 251: * .. External Functions .. 252: LOGICAL LSAME 253: DOUBLE PRECISION DLAMCH, ZLANHB 254: EXTERNAL LSAME, DLAMCH, ZLANHB 255: * .. 256: * .. External Subroutines .. 257: EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAQHB, ZPBCON, ZPBEQU, 258: $ ZPBRFS, ZPBTRF, ZPBTRS 259: * .. 260: * .. Intrinsic Functions .. 261: INTRINSIC MAX, MIN 262: * .. 263: * .. Executable Statements .. 264: * 265: INFO = 0 266: NOFACT = LSAME( FACT, 'N' ) 267: EQUIL = LSAME( FACT, 'E' ) 268: UPPER = LSAME( UPLO, 'U' ) 269: IF( NOFACT .OR. EQUIL ) THEN 270: EQUED = 'N' 271: RCEQU = .FALSE. 272: ELSE 273: RCEQU = LSAME( EQUED, 'Y' ) 274: SMLNUM = DLAMCH( 'Safe minimum' ) 275: BIGNUM = ONE / SMLNUM 276: END IF 277: * 278: * Test the input parameters. 279: * 280: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) 281: $ THEN 282: INFO = -1 283: ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 284: INFO = -2 285: ELSE IF( N.LT.0 ) THEN 286: INFO = -3 287: ELSE IF( KD.LT.0 ) THEN 288: INFO = -4 289: ELSE IF( NRHS.LT.0 ) THEN 290: INFO = -5 291: ELSE IF( LDAB.LT.KD+1 ) THEN 292: INFO = -7 293: ELSE IF( LDAFB.LT.KD+1 ) THEN 294: INFO = -9 295: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 296: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 297: INFO = -10 298: ELSE 299: IF( RCEQU ) THEN 300: SMIN = BIGNUM 301: SMAX = ZERO 302: DO 10 J = 1, N 303: SMIN = MIN( SMIN, S( J ) ) 304: SMAX = MAX( SMAX, S( J ) ) 305: 10 CONTINUE 306: IF( SMIN.LE.ZERO ) THEN 307: INFO = -11 308: ELSE IF( N.GT.0 ) THEN 309: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 310: ELSE 311: SCOND = ONE 312: END IF 313: END IF 314: IF( INFO.EQ.0 ) THEN 315: IF( LDB.LT.MAX( 1, N ) ) THEN 316: INFO = -13 317: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 318: INFO = -15 319: END IF 320: END IF 321: END IF 322: * 323: IF( INFO.NE.0 ) THEN 324: CALL XERBLA( 'ZPBSVX', -INFO ) 325: RETURN 326: END IF 327: * 328: IF( EQUIL ) THEN 329: * 330: * Compute row and column scalings to equilibrate the matrix A. 331: * 332: CALL ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU ) 333: IF( INFEQU.EQ.0 ) THEN 334: * 335: * Equilibrate the matrix. 336: * 337: CALL ZLAQHB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED ) 338: RCEQU = LSAME( EQUED, 'Y' ) 339: END IF 340: END IF 341: * 342: * Scale the right-hand side. 343: * 344: IF( RCEQU ) THEN 345: DO 30 J = 1, NRHS 346: DO 20 I = 1, N 347: B( I, J ) = S( I )*B( I, J ) 348: 20 CONTINUE 349: 30 CONTINUE 350: END IF 351: * 352: IF( NOFACT .OR. EQUIL ) THEN 353: * 354: * Compute the Cholesky factorization A = U'*U or A = L*L'. 355: * 356: IF( UPPER ) THEN 357: DO 40 J = 1, N 358: J1 = MAX( J-KD, 1 ) 359: CALL ZCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1, 360: $ AFB( KD+1-J+J1, J ), 1 ) 361: 40 CONTINUE 362: ELSE 363: DO 50 J = 1, N 364: J2 = MIN( J+KD, N ) 365: CALL ZCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 ) 366: 50 CONTINUE 367: END IF 368: * 369: CALL ZPBTRF( UPLO, N, KD, AFB, LDAFB, INFO ) 370: * 371: * Return if INFO is non-zero. 372: * 373: IF( INFO.GT.0 )THEN 374: RCOND = ZERO 375: RETURN 376: END IF 377: END IF 378: * 379: * Compute the norm of the matrix A. 380: * 381: ANORM = ZLANHB( '1', UPLO, N, KD, AB, LDAB, RWORK ) 382: * 383: * Compute the reciprocal of the condition number of A. 384: * 385: CALL ZPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, RWORK, 386: $ INFO ) 387: * 388: * Compute the solution matrix X. 389: * 390: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 391: CALL ZPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO ) 392: * 393: * Use iterative refinement to improve the computed solution and 394: * compute error bounds and backward error estimates for it. 395: * 396: CALL ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, 397: $ LDX, FERR, BERR, WORK, RWORK, INFO ) 398: * 399: * Transform the solution matrix X to a solution of the original 400: * system. 401: * 402: IF( RCEQU ) THEN 403: DO 70 J = 1, NRHS 404: DO 60 I = 1, N 405: X( I, J ) = S( I )*X( I, J ) 406: 60 CONTINUE 407: 70 CONTINUE 408: DO 80 J = 1, NRHS 409: FERR( J ) = FERR( J ) / SCOND 410: 80 CONTINUE 411: END IF 412: * 413: * Set INFO = N+1 if the matrix is singular to working precision. 414: * 415: IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) 416: $ INFO = N + 1 417: * 418: RETURN 419: * 420: * End of ZPBSVX 421: * 422: END