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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, 2: $ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, 3: $ WORK, IWORK, 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: INTEGER IWORK( * ) 17: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 18: $ BERR( * ), FERR( * ), S( * ), WORK( * ), 19: $ X( LDX, * ) 20: * .. 21: * 22: * Purpose 23: * ======= 24: * 25: * DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to 26: * compute the solution to a real system of linear equations 27: * A * X = B, 28: * where A is an N-by-N symmetric positive definite band matrix and X 29: * and B are N-by-NRHS matrices. 30: * 31: * Error bounds on the solution and a condition estimate are also 32: * provided. 33: * 34: * Description 35: * =========== 36: * 37: * The following steps are performed: 38: * 39: * 1. If FACT = 'E', real scaling factors are computed to equilibrate 40: * the system: 41: * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B 42: * Whether or not the system will be equilibrated depends on the 43: * scaling of the matrix A, but if equilibration is used, A is 44: * overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 45: * 46: * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to 47: * factor the matrix A (after equilibration if FACT = 'E') as 48: * A = U**T * U, if UPLO = 'U', or 49: * A = L * L**T, if UPLO = 'L', 50: * where U is an upper triangular band matrix, and L is a lower 51: * triangular band matrix. 52: * 53: * 3. If the leading i-by-i principal minor is not positive definite, 54: * then the routine returns with INFO = i. Otherwise, the factored 55: * form of A is used to estimate the condition number of the matrix 56: * A. If the reciprocal of the condition number is less than machine 57: * precision, INFO = N+1 is returned as a warning, but the routine 58: * still goes on to solve for X and compute error bounds as 59: * described below. 60: * 61: * 4. The system of equations is solved for X using the factored form 62: * of A. 63: * 64: * 5. Iterative refinement is applied to improve the computed solution 65: * matrix and calculate error bounds and backward error estimates 66: * for it. 67: * 68: * 6. If equilibration was used, the matrix X is premultiplied by 69: * diag(S) so that it solves the original system before 70: * equilibration. 71: * 72: * Arguments 73: * ========= 74: * 75: * FACT (input) CHARACTER*1 76: * Specifies whether or not the factored form of the matrix A is 77: * supplied on entry, and if not, whether the matrix A should be 78: * equilibrated before it is factored. 79: * = 'F': On entry, AFB contains the factored form of A. 80: * If EQUED = 'Y', the matrix A has been equilibrated 81: * with scaling factors given by S. AB and AFB will not 82: * be modified. 83: * = 'N': The matrix A will be copied to AFB and factored. 84: * = 'E': The matrix A will be equilibrated if necessary, then 85: * copied to AFB and factored. 86: * 87: * UPLO (input) CHARACTER*1 88: * = 'U': Upper triangle of A is stored; 89: * = 'L': Lower triangle of A is stored. 90: * 91: * N (input) INTEGER 92: * The number of linear equations, i.e., the order of the 93: * matrix A. N >= 0. 94: * 95: * KD (input) INTEGER 96: * The number of superdiagonals of the matrix A if UPLO = 'U', 97: * or the number of subdiagonals if UPLO = 'L'. KD >= 0. 98: * 99: * NRHS (input) INTEGER 100: * The number of right-hand sides, i.e., the number of columns 101: * of the matrices B and X. NRHS >= 0. 102: * 103: * AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) 104: * On entry, the upper or lower triangle of the symmetric band 105: * matrix A, stored in the first KD+1 rows of the array, except 106: * if FACT = 'F' and EQUED = 'Y', then A must contain the 107: * equilibrated matrix diag(S)*A*diag(S). The j-th column of A 108: * is stored in the j-th column of the array AB as follows: 109: * if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j; 110: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD). 111: * See below for further details. 112: * 113: * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 114: * diag(S)*A*diag(S). 115: * 116: * LDAB (input) INTEGER 117: * The leading dimension of the array A. LDAB >= KD+1. 118: * 119: * AFB (input or output) DOUBLE PRECISION array, dimension (LDAFB,N) 120: * If FACT = 'F', then AFB is an input argument and on entry 121: * contains the triangular factor U or L from the Cholesky 122: * factorization A = U**T*U or A = L*L**T of the band matrix 123: * A, in the same storage format as A (see AB). If EQUED = 'Y', 124: * then AFB is the factored form of the equilibrated matrix A. 125: * 126: * If FACT = 'N', then AFB is an output argument and on exit 127: * returns the triangular factor U or L from the Cholesky 128: * factorization A = U**T*U or A = L*L**T. 129: * 130: * If FACT = 'E', then AFB is an output argument and on exit 131: * returns the triangular factor U or L from the Cholesky 132: * factorization A = U**T*U or A = L*L**T of the equilibrated 133: * matrix A (see the description of A for the form of the 134: * equilibrated matrix). 135: * 136: * LDAFB (input) INTEGER 137: * The leading dimension of the array AFB. LDAFB >= KD+1. 138: * 139: * EQUED (input or output) CHARACTER*1 140: * Specifies the form of equilibration that was done. 141: * = 'N': No equilibration (always true if FACT = 'N'). 142: * = 'Y': Equilibration was done, i.e., A has been replaced by 143: * diag(S) * A * diag(S). 144: * EQUED is an input argument if FACT = 'F'; otherwise, it is an 145: * output argument. 146: * 147: * S (input or output) DOUBLE PRECISION array, dimension (N) 148: * The scale factors for A; not accessed if EQUED = 'N'. S is 149: * an input argument if FACT = 'F'; otherwise, S is an output 150: * argument. If FACT = 'F' and EQUED = 'Y', each element of S 151: * must be positive. 152: * 153: * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) 154: * On entry, the N-by-NRHS right hand side matrix B. 155: * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', 156: * B is overwritten by diag(S) * B. 157: * 158: * LDB (input) INTEGER 159: * The leading dimension of the array B. LDB >= max(1,N). 160: * 161: * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) 162: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to 163: * the original system of equations. Note that if EQUED = 'Y', 164: * A and B are modified on exit, and the solution to the 165: * equilibrated system is inv(diag(S))*X. 166: * 167: * LDX (input) INTEGER 168: * The leading dimension of the array X. LDX >= max(1,N). 169: * 170: * RCOND (output) DOUBLE PRECISION 171: * The estimate of the reciprocal condition number of the matrix 172: * A after equilibration (if done). If RCOND is less than the 173: * machine precision (in particular, if RCOND = 0), the matrix 174: * is singular to working precision. This condition is 175: * indicated by a return code of INFO > 0. 176: * 177: * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 178: * The estimated forward error bound for each solution vector 179: * X(j) (the j-th column of the solution matrix X). 180: * If XTRUE is the true solution corresponding to X(j), FERR(j) 181: * is an estimated upper bound for the magnitude of the largest 182: * element in (X(j) - XTRUE) divided by the magnitude of the 183: * largest element in X(j). The estimate is as reliable as 184: * the estimate for RCOND, and is almost always a slight 185: * overestimate of the true error. 186: * 187: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 188: * The componentwise relative backward error of each solution 189: * vector X(j) (i.e., the smallest relative change in 190: * any element of A or B that makes X(j) an exact solution). 191: * 192: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) 193: * 194: * IWORK (workspace) INTEGER array, dimension (N) 195: * 196: * INFO (output) INTEGER 197: * = 0: successful exit 198: * < 0: if INFO = -i, the i-th argument had an illegal value 199: * > 0: if INFO = i, and i is 200: * <= N: the leading minor of order i of A is 201: * not positive definite, so the factorization 202: * could not be completed, and the solution has not 203: * been computed. RCOND = 0 is returned. 204: * = N+1: U is nonsingular, but RCOND is less than machine 205: * precision, meaning that the matrix is singular 206: * to working precision. Nevertheless, the 207: * solution and error bounds are computed because 208: * there are a number of situations where the 209: * computed solution can be more accurate than the 210: * value of RCOND would suggest. 211: * 212: * Further Details 213: * =============== 214: * 215: * The band storage scheme is illustrated by the following example, when 216: * N = 6, KD = 2, and UPLO = 'U': 217: * 218: * Two-dimensional storage of the symmetric matrix A: 219: * 220: * a11 a12 a13 221: * a22 a23 a24 222: * a33 a34 a35 223: * a44 a45 a46 224: * a55 a56 225: * (aij=conjg(aji)) a66 226: * 227: * Band storage of the upper triangle of A: 228: * 229: * * * a13 a24 a35 a46 230: * * a12 a23 a34 a45 a56 231: * a11 a22 a33 a44 a55 a66 232: * 233: * Similarly, if UPLO = 'L' the format of A is as follows: 234: * 235: * a11 a22 a33 a44 a55 a66 236: * a21 a32 a43 a54 a65 * 237: * a31 a42 a53 a64 * * 238: * 239: * Array elements marked * are not used by the routine. 240: * 241: * ===================================================================== 242: * 243: * .. Parameters .. 244: DOUBLE PRECISION ZERO, ONE 245: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 246: * .. 247: * .. Local Scalars .. 248: LOGICAL EQUIL, NOFACT, RCEQU, UPPER 249: INTEGER I, INFEQU, J, J1, J2 250: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM 251: * .. 252: * .. External Functions .. 253: LOGICAL LSAME 254: DOUBLE PRECISION DLAMCH, DLANSB 255: EXTERNAL LSAME, DLAMCH, DLANSB 256: * .. 257: * .. External Subroutines .. 258: EXTERNAL DCOPY, DLACPY, DLAQSB, DPBCON, DPBEQU, DPBRFS, 259: $ DPBTRF, DPBTRS, XERBLA 260: * .. 261: * .. Intrinsic Functions .. 262: INTRINSIC MAX, MIN 263: * .. 264: * .. Executable Statements .. 265: * 266: INFO = 0 267: NOFACT = LSAME( FACT, 'N' ) 268: EQUIL = LSAME( FACT, 'E' ) 269: UPPER = LSAME( UPLO, 'U' ) 270: IF( NOFACT .OR. EQUIL ) THEN 271: EQUED = 'N' 272: RCEQU = .FALSE. 273: ELSE 274: RCEQU = LSAME( EQUED, 'Y' ) 275: SMLNUM = DLAMCH( 'Safe minimum' ) 276: BIGNUM = ONE / SMLNUM 277: END IF 278: * 279: * Test the input parameters. 280: * 281: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) 282: $ THEN 283: INFO = -1 284: ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 285: INFO = -2 286: ELSE IF( N.LT.0 ) THEN 287: INFO = -3 288: ELSE IF( KD.LT.0 ) THEN 289: INFO = -4 290: ELSE IF( NRHS.LT.0 ) THEN 291: INFO = -5 292: ELSE IF( LDAB.LT.KD+1 ) THEN 293: INFO = -7 294: ELSE IF( LDAFB.LT.KD+1 ) THEN 295: INFO = -9 296: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 297: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 298: INFO = -10 299: ELSE 300: IF( RCEQU ) THEN 301: SMIN = BIGNUM 302: SMAX = ZERO 303: DO 10 J = 1, N 304: SMIN = MIN( SMIN, S( J ) ) 305: SMAX = MAX( SMAX, S( J ) ) 306: 10 CONTINUE 307: IF( SMIN.LE.ZERO ) THEN 308: INFO = -11 309: ELSE IF( N.GT.0 ) THEN 310: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 311: ELSE 312: SCOND = ONE 313: END IF 314: END IF 315: IF( INFO.EQ.0 ) THEN 316: IF( LDB.LT.MAX( 1, N ) ) THEN 317: INFO = -13 318: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 319: INFO = -15 320: END IF 321: END IF 322: END IF 323: * 324: IF( INFO.NE.0 ) THEN 325: CALL XERBLA( 'DPBSVX', -INFO ) 326: RETURN 327: END IF 328: * 329: IF( EQUIL ) THEN 330: * 331: * Compute row and column scalings to equilibrate the matrix A. 332: * 333: CALL DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU ) 334: IF( INFEQU.EQ.0 ) THEN 335: * 336: * Equilibrate the matrix. 337: * 338: CALL DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED ) 339: RCEQU = LSAME( EQUED, 'Y' ) 340: END IF 341: END IF 342: * 343: * Scale the right-hand side. 344: * 345: IF( RCEQU ) THEN 346: DO 30 J = 1, NRHS 347: DO 20 I = 1, N 348: B( I, J ) = S( I )*B( I, J ) 349: 20 CONTINUE 350: 30 CONTINUE 351: END IF 352: * 353: IF( NOFACT .OR. EQUIL ) THEN 354: * 355: * Compute the Cholesky factorization A = U'*U or A = L*L'. 356: * 357: IF( UPPER ) THEN 358: DO 40 J = 1, N 359: J1 = MAX( J-KD, 1 ) 360: CALL DCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1, 361: $ AFB( KD+1-J+J1, J ), 1 ) 362: 40 CONTINUE 363: ELSE 364: DO 50 J = 1, N 365: J2 = MIN( J+KD, N ) 366: CALL DCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 ) 367: 50 CONTINUE 368: END IF 369: * 370: CALL DPBTRF( UPLO, N, KD, AFB, LDAFB, INFO ) 371: * 372: * Return if INFO is non-zero. 373: * 374: IF( INFO.GT.0 )THEN 375: RCOND = ZERO 376: RETURN 377: END IF 378: END IF 379: * 380: * Compute the norm of the matrix A. 381: * 382: ANORM = DLANSB( '1', UPLO, N, KD, AB, LDAB, WORK ) 383: * 384: * Compute the reciprocal of the condition number of A. 385: * 386: CALL DPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, IWORK, 387: $ INFO ) 388: * 389: * Compute the solution matrix X. 390: * 391: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 392: CALL DPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO ) 393: * 394: * Use iterative refinement to improve the computed solution and 395: * compute error bounds and backward error estimates for it. 396: * 397: CALL DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, 398: $ LDX, FERR, BERR, WORK, IWORK, INFO ) 399: * 400: * Transform the solution matrix X to a solution of the original 401: * system. 402: * 403: IF( RCEQU ) THEN 404: DO 70 J = 1, NRHS 405: DO 60 I = 1, N 406: X( I, J ) = S( I )*X( I, J ) 407: 60 CONTINUE 408: 70 CONTINUE 409: DO 80 J = 1, NRHS 410: FERR( J ) = FERR( J ) / SCOND 411: 80 CONTINUE 412: END IF 413: * 414: * Set INFO = N+1 if the matrix is singular to working precision. 415: * 416: IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) 417: $ INFO = N + 1 418: * 419: RETURN 420: * 421: * End of DPBSVX 422: * 423: END