Annotation of rpl/lapack/lapack/dposvx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
! 2: $ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
! 3: $ 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, LDA, LDAF, LDB, LDX, N, NRHS
! 13: DOUBLE PRECISION RCOND
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IWORK( * )
! 17: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
! 18: $ BERR( * ), FERR( * ), S( * ), WORK( * ),
! 19: $ X( LDX, * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * DPOSVX 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 matrix and X and B
! 29: * 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 matrix and L is a lower triangular
! 51: * 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, AF contains the factored form of A.
! 80: * If EQUED = 'Y', the matrix A has been equilibrated
! 81: * with scaling factors given by S. A and AF will not
! 82: * be modified.
! 83: * = 'N': The matrix A will be copied to AF and factored.
! 84: * = 'E': The matrix A will be equilibrated if necessary, then
! 85: * copied to AF 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: * NRHS (input) INTEGER
! 96: * The number of right hand sides, i.e., the number of columns
! 97: * of the matrices B and X. NRHS >= 0.
! 98: *
! 99: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
! 100: * On entry, the symmetric matrix A, except if FACT = 'F' and
! 101: * EQUED = 'Y', then A must contain the equilibrated matrix
! 102: * diag(S)*A*diag(S). If UPLO = 'U', the leading
! 103: * N-by-N upper triangular part of A contains the upper
! 104: * triangular part of the matrix A, and the strictly lower
! 105: * triangular part of A is not referenced. If UPLO = 'L', the
! 106: * leading N-by-N lower triangular part of A contains the lower
! 107: * triangular part of the matrix A, and the strictly upper
! 108: * triangular part of A is not referenced. A is not modified if
! 109: * FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
! 110: *
! 111: * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
! 112: * diag(S)*A*diag(S).
! 113: *
! 114: * LDA (input) INTEGER
! 115: * The leading dimension of the array A. LDA >= max(1,N).
! 116: *
! 117: * AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
! 118: * If FACT = 'F', then AF is an input argument and on entry
! 119: * contains the triangular factor U or L from the Cholesky
! 120: * factorization A = U**T*U or A = L*L**T, in the same storage
! 121: * format as A. If EQUED .ne. 'N', then AF is the factored form
! 122: * of the equilibrated matrix diag(S)*A*diag(S).
! 123: *
! 124: * If FACT = 'N', then AF is an output argument and on exit
! 125: * returns the triangular factor U or L from the Cholesky
! 126: * factorization A = U**T*U or A = L*L**T of the original
! 127: * matrix A.
! 128: *
! 129: * If FACT = 'E', then AF is an output argument and on exit
! 130: * returns the triangular factor U or L from the Cholesky
! 131: * factorization A = U**T*U or A = L*L**T of the equilibrated
! 132: * matrix A (see the description of A for the form of the
! 133: * equilibrated matrix).
! 134: *
! 135: * LDAF (input) INTEGER
! 136: * The leading dimension of the array AF. LDAF >= max(1,N).
! 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (3*N)
! 192: *
! 193: * IWORK (workspace) INTEGER 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: * =====================================================================
! 212: *
! 213: * .. Parameters ..
! 214: DOUBLE PRECISION ZERO, ONE
! 215: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 216: * ..
! 217: * .. Local Scalars ..
! 218: LOGICAL EQUIL, NOFACT, RCEQU
! 219: INTEGER I, INFEQU, J
! 220: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
! 221: * ..
! 222: * .. External Functions ..
! 223: LOGICAL LSAME
! 224: DOUBLE PRECISION DLAMCH, DLANSY
! 225: EXTERNAL LSAME, DLAMCH, DLANSY
! 226: * ..
! 227: * .. External Subroutines ..
! 228: EXTERNAL DLACPY, DLAQSY, DPOCON, DPOEQU, DPORFS, DPOTRF,
! 229: $ DPOTRS, XERBLA
! 230: * ..
! 231: * .. Intrinsic Functions ..
! 232: INTRINSIC MAX, MIN
! 233: * ..
! 234: * .. Executable Statements ..
! 235: *
! 236: INFO = 0
! 237: NOFACT = LSAME( FACT, 'N' )
! 238: EQUIL = LSAME( FACT, 'E' )
! 239: IF( NOFACT .OR. EQUIL ) THEN
! 240: EQUED = 'N'
! 241: RCEQU = .FALSE.
! 242: ELSE
! 243: RCEQU = LSAME( EQUED, 'Y' )
! 244: SMLNUM = DLAMCH( 'Safe minimum' )
! 245: BIGNUM = ONE / SMLNUM
! 246: END IF
! 247: *
! 248: * Test the input parameters.
! 249: *
! 250: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
! 251: $ THEN
! 252: INFO = -1
! 253: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
! 254: $ THEN
! 255: INFO = -2
! 256: ELSE IF( N.LT.0 ) THEN
! 257: INFO = -3
! 258: ELSE IF( NRHS.LT.0 ) THEN
! 259: INFO = -4
! 260: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 261: INFO = -6
! 262: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
! 263: INFO = -8
! 264: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
! 265: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
! 266: INFO = -9
! 267: ELSE
! 268: IF( RCEQU ) THEN
! 269: SMIN = BIGNUM
! 270: SMAX = ZERO
! 271: DO 10 J = 1, N
! 272: SMIN = MIN( SMIN, S( J ) )
! 273: SMAX = MAX( SMAX, S( J ) )
! 274: 10 CONTINUE
! 275: IF( SMIN.LE.ZERO ) THEN
! 276: INFO = -10
! 277: ELSE IF( N.GT.0 ) THEN
! 278: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
! 279: ELSE
! 280: SCOND = ONE
! 281: END IF
! 282: END IF
! 283: IF( INFO.EQ.0 ) THEN
! 284: IF( LDB.LT.MAX( 1, N ) ) THEN
! 285: INFO = -12
! 286: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 287: INFO = -14
! 288: END IF
! 289: END IF
! 290: END IF
! 291: *
! 292: IF( INFO.NE.0 ) THEN
! 293: CALL XERBLA( 'DPOSVX', -INFO )
! 294: RETURN
! 295: END IF
! 296: *
! 297: IF( EQUIL ) THEN
! 298: *
! 299: * Compute row and column scalings to equilibrate the matrix A.
! 300: *
! 301: CALL DPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU )
! 302: IF( INFEQU.EQ.0 ) THEN
! 303: *
! 304: * Equilibrate the matrix.
! 305: *
! 306: CALL DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
! 307: RCEQU = LSAME( EQUED, 'Y' )
! 308: END IF
! 309: END IF
! 310: *
! 311: * Scale the right hand side.
! 312: *
! 313: IF( RCEQU ) THEN
! 314: DO 30 J = 1, NRHS
! 315: DO 20 I = 1, N
! 316: B( I, J ) = S( I )*B( I, J )
! 317: 20 CONTINUE
! 318: 30 CONTINUE
! 319: END IF
! 320: *
! 321: IF( NOFACT .OR. EQUIL ) THEN
! 322: *
! 323: * Compute the Cholesky factorization A = U'*U or A = L*L'.
! 324: *
! 325: CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
! 326: CALL DPOTRF( UPLO, N, AF, LDAF, INFO )
! 327: *
! 328: * Return if INFO is non-zero.
! 329: *
! 330: IF( INFO.GT.0 )THEN
! 331: RCOND = ZERO
! 332: RETURN
! 333: END IF
! 334: END IF
! 335: *
! 336: * Compute the norm of the matrix A.
! 337: *
! 338: ANORM = DLANSY( '1', UPLO, N, A, LDA, WORK )
! 339: *
! 340: * Compute the reciprocal of the condition number of A.
! 341: *
! 342: CALL DPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO )
! 343: *
! 344: * Compute the solution matrix X.
! 345: *
! 346: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
! 347: CALL DPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
! 348: *
! 349: * Use iterative refinement to improve the computed solution and
! 350: * compute error bounds and backward error estimates for it.
! 351: *
! 352: CALL DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX,
! 353: $ FERR, BERR, WORK, IWORK, INFO )
! 354: *
! 355: * Transform the solution matrix X to a solution of the original
! 356: * system.
! 357: *
! 358: IF( RCEQU ) THEN
! 359: DO 50 J = 1, NRHS
! 360: DO 40 I = 1, N
! 361: X( I, J ) = S( I )*X( I, J )
! 362: 40 CONTINUE
! 363: 50 CONTINUE
! 364: DO 60 J = 1, NRHS
! 365: FERR( J ) = FERR( J ) / SCOND
! 366: 60 CONTINUE
! 367: END IF
! 368: *
! 369: * Set INFO = N+1 if the matrix is singular to working precision.
! 370: *
! 371: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
! 372: $ INFO = N + 1
! 373: *
! 374: RETURN
! 375: *
! 376: * End of DPOSVX
! 377: *
! 378: END
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