Annotation of rpl/lapack/lapack/zppsvx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
! 2: $ X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
! 3: *
! 4: * -- LAPACK driver routine (version 3.2) --
! 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 7: * November 2006
! 8: *
! 9: * .. Scalar Arguments ..
! 10: CHARACTER EQUED, FACT, UPLO
! 11: INTEGER INFO, LDB, LDX, N, NRHS
! 12: DOUBLE PRECISION RCOND
! 13: * ..
! 14: * .. Array Arguments ..
! 15: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
! 16: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
! 17: $ X( LDX, * )
! 18: * ..
! 19: *
! 20: * Purpose
! 21: * =======
! 22: *
! 23: * ZPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
! 24: * compute the solution to a complex system of linear equations
! 25: * A * X = B,
! 26: * where A is an N-by-N Hermitian positive definite matrix stored in
! 27: * packed format and X and B are N-by-NRHS matrices.
! 28: *
! 29: * Error bounds on the solution and a condition estimate are also
! 30: * provided.
! 31: *
! 32: * Description
! 33: * ===========
! 34: *
! 35: * The following steps are performed:
! 36: *
! 37: * 1. If FACT = 'E', real scaling factors are computed to equilibrate
! 38: * the system:
! 39: * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
! 40: * Whether or not the system will be equilibrated depends on the
! 41: * scaling of the matrix A, but if equilibration is used, A is
! 42: * overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
! 43: *
! 44: * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
! 45: * factor the matrix A (after equilibration if FACT = 'E') as
! 46: * A = U'* U , if UPLO = 'U', or
! 47: * A = L * L', if UPLO = 'L',
! 48: * where U is an upper triangular matrix, L is a lower triangular
! 49: * matrix, and ' indicates conjugate transpose.
! 50: *
! 51: * 3. If the leading i-by-i principal minor is not positive definite,
! 52: * then the routine returns with INFO = i. Otherwise, the factored
! 53: * form of A is used to estimate the condition number of the matrix
! 54: * A. If the reciprocal of the condition number is less than machine
! 55: * precision, INFO = N+1 is returned as a warning, but the routine
! 56: * still goes on to solve for X and compute error bounds as
! 57: * described below.
! 58: *
! 59: * 4. The system of equations is solved for X using the factored form
! 60: * of A.
! 61: *
! 62: * 5. Iterative refinement is applied to improve the computed solution
! 63: * matrix and calculate error bounds and backward error estimates
! 64: * for it.
! 65: *
! 66: * 6. If equilibration was used, the matrix X is premultiplied by
! 67: * diag(S) so that it solves the original system before
! 68: * equilibration.
! 69: *
! 70: * Arguments
! 71: * =========
! 72: *
! 73: * FACT (input) CHARACTER*1
! 74: * Specifies whether or not the factored form of the matrix A is
! 75: * supplied on entry, and if not, whether the matrix A should be
! 76: * equilibrated before it is factored.
! 77: * = 'F': On entry, AFP contains the factored form of A.
! 78: * If EQUED = 'Y', the matrix A has been equilibrated
! 79: * with scaling factors given by S. AP and AFP will not
! 80: * be modified.
! 81: * = 'N': The matrix A will be copied to AFP and factored.
! 82: * = 'E': The matrix A will be equilibrated if necessary, then
! 83: * copied to AFP and factored.
! 84: *
! 85: * UPLO (input) CHARACTER*1
! 86: * = 'U': Upper triangle of A is stored;
! 87: * = 'L': Lower triangle of A is stored.
! 88: *
! 89: * N (input) INTEGER
! 90: * The number of linear equations, i.e., the order of the
! 91: * matrix A. N >= 0.
! 92: *
! 93: * NRHS (input) INTEGER
! 94: * The number of right hand sides, i.e., the number of columns
! 95: * of the matrices B and X. NRHS >= 0.
! 96: *
! 97: * AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
! 98: * On entry, the upper or lower triangle of the Hermitian matrix
! 99: * A, packed columnwise in a linear array, except if FACT = 'F'
! 100: * and EQUED = 'Y', then A must contain the equilibrated matrix
! 101: * diag(S)*A*diag(S). The j-th column of A is stored in the
! 102: * array AP as follows:
! 103: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 104: * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
! 105: * See below for further details. A is not modified if
! 106: * FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
! 107: *
! 108: * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
! 109: * diag(S)*A*diag(S).
! 110: *
! 111: * AFP (input or output) COMPLEX*16 array, dimension (N*(N+1)/2)
! 112: * If FACT = 'F', then AFP is an input argument and on entry
! 113: * contains the triangular factor U or L from the Cholesky
! 114: * factorization A = U**H*U or A = L*L**H, in the same storage
! 115: * format as A. If EQUED .ne. 'N', then AFP is the factored
! 116: * form of the equilibrated matrix A.
! 117: *
! 118: * If FACT = 'N', then AFP is an output argument and on exit
! 119: * returns the triangular factor U or L from the Cholesky
! 120: * factorization A = U**H*U or A = L*L**H of the original
! 121: * matrix A.
! 122: *
! 123: * If FACT = 'E', then AFP is an output argument and on exit
! 124: * returns the triangular factor U or L from the Cholesky
! 125: * factorization A = U**H*U or A = L*L**H of the equilibrated
! 126: * matrix A (see the description of AP for the form of the
! 127: * equilibrated matrix).
! 128: *
! 129: * EQUED (input or output) CHARACTER*1
! 130: * Specifies the form of equilibration that was done.
! 131: * = 'N': No equilibration (always true if FACT = 'N').
! 132: * = 'Y': Equilibration was done, i.e., A has been replaced by
! 133: * diag(S) * A * diag(S).
! 134: * EQUED is an input argument if FACT = 'F'; otherwise, it is an
! 135: * output argument.
! 136: *
! 137: * S (input or output) DOUBLE PRECISION array, dimension (N)
! 138: * The scale factors for A; not accessed if EQUED = 'N'. S is
! 139: * an input argument if FACT = 'F'; otherwise, S is an output
! 140: * argument. If FACT = 'F' and EQUED = 'Y', each element of S
! 141: * must be positive.
! 142: *
! 143: * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
! 144: * On entry, the N-by-NRHS right hand side matrix B.
! 145: * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
! 146: * B is overwritten by diag(S) * B.
! 147: *
! 148: * LDB (input) INTEGER
! 149: * The leading dimension of the array B. LDB >= max(1,N).
! 150: *
! 151: * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
! 152: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
! 153: * the original system of equations. Note that if EQUED = 'Y',
! 154: * A and B are modified on exit, and the solution to the
! 155: * equilibrated system is inv(diag(S))*X.
! 156: *
! 157: * LDX (input) INTEGER
! 158: * The leading dimension of the array X. LDX >= max(1,N).
! 159: *
! 160: * RCOND (output) DOUBLE PRECISION
! 161: * The estimate of the reciprocal condition number of the matrix
! 162: * A after equilibration (if done). If RCOND is less than the
! 163: * machine precision (in particular, if RCOND = 0), the matrix
! 164: * is singular to working precision. This condition is
! 165: * indicated by a return code of INFO > 0.
! 166: *
! 167: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 168: * The estimated forward error bound for each solution vector
! 169: * X(j) (the j-th column of the solution matrix X).
! 170: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 171: * is an estimated upper bound for the magnitude of the largest
! 172: * element in (X(j) - XTRUE) divided by the magnitude of the
! 173: * largest element in X(j). The estimate is as reliable as
! 174: * the estimate for RCOND, and is almost always a slight
! 175: * overestimate of the true error.
! 176: *
! 177: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 178: * The componentwise relative backward error of each solution
! 179: * vector X(j) (i.e., the smallest relative change in
! 180: * any element of A or B that makes X(j) an exact solution).
! 181: *
! 182: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
! 183: *
! 184: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
! 185: *
! 186: * INFO (output) INTEGER
! 187: * = 0: successful exit
! 188: * < 0: if INFO = -i, the i-th argument had an illegal value
! 189: * > 0: if INFO = i, and i is
! 190: * <= N: the leading minor of order i of A is
! 191: * not positive definite, so the factorization
! 192: * could not be completed, and the solution has not
! 193: * been computed. RCOND = 0 is returned.
! 194: * = N+1: U is nonsingular, but RCOND is less than machine
! 195: * precision, meaning that the matrix is singular
! 196: * to working precision. Nevertheless, the
! 197: * solution and error bounds are computed because
! 198: * there are a number of situations where the
! 199: * computed solution can be more accurate than the
! 200: * value of RCOND would suggest.
! 201: *
! 202: * Further Details
! 203: * ===============
! 204: *
! 205: * The packed storage scheme is illustrated by the following example
! 206: * when N = 4, UPLO = 'U':
! 207: *
! 208: * Two-dimensional storage of the Hermitian matrix A:
! 209: *
! 210: * a11 a12 a13 a14
! 211: * a22 a23 a24
! 212: * a33 a34 (aij = conjg(aji))
! 213: * a44
! 214: *
! 215: * Packed storage of the upper triangle of A:
! 216: *
! 217: * AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
! 218: *
! 219: * =====================================================================
! 220: *
! 221: * .. Parameters ..
! 222: DOUBLE PRECISION ZERO, ONE
! 223: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 224: * ..
! 225: * .. Local Scalars ..
! 226: LOGICAL EQUIL, NOFACT, RCEQU
! 227: INTEGER I, INFEQU, J
! 228: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
! 229: * ..
! 230: * .. External Functions ..
! 231: LOGICAL LSAME
! 232: DOUBLE PRECISION DLAMCH, ZLANHP
! 233: EXTERNAL LSAME, DLAMCH, ZLANHP
! 234: * ..
! 235: * .. External Subroutines ..
! 236: EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAQHP, ZPPCON, ZPPEQU,
! 237: $ ZPPRFS, ZPPTRF, ZPPTRS
! 238: * ..
! 239: * .. Intrinsic Functions ..
! 240: INTRINSIC MAX, MIN
! 241: * ..
! 242: * .. Executable Statements ..
! 243: *
! 244: INFO = 0
! 245: NOFACT = LSAME( FACT, 'N' )
! 246: EQUIL = LSAME( FACT, 'E' )
! 247: IF( NOFACT .OR. EQUIL ) THEN
! 248: EQUED = 'N'
! 249: RCEQU = .FALSE.
! 250: ELSE
! 251: RCEQU = LSAME( EQUED, 'Y' )
! 252: SMLNUM = DLAMCH( 'Safe minimum' )
! 253: BIGNUM = ONE / SMLNUM
! 254: END IF
! 255: *
! 256: * Test the input parameters.
! 257: *
! 258: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
! 259: $ THEN
! 260: INFO = -1
! 261: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
! 262: $ THEN
! 263: INFO = -2
! 264: ELSE IF( N.LT.0 ) THEN
! 265: INFO = -3
! 266: ELSE IF( NRHS.LT.0 ) THEN
! 267: INFO = -4
! 268: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
! 269: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
! 270: INFO = -7
! 271: ELSE
! 272: IF( RCEQU ) THEN
! 273: SMIN = BIGNUM
! 274: SMAX = ZERO
! 275: DO 10 J = 1, N
! 276: SMIN = MIN( SMIN, S( J ) )
! 277: SMAX = MAX( SMAX, S( J ) )
! 278: 10 CONTINUE
! 279: IF( SMIN.LE.ZERO ) THEN
! 280: INFO = -8
! 281: ELSE IF( N.GT.0 ) THEN
! 282: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
! 283: ELSE
! 284: SCOND = ONE
! 285: END IF
! 286: END IF
! 287: IF( INFO.EQ.0 ) THEN
! 288: IF( LDB.LT.MAX( 1, N ) ) THEN
! 289: INFO = -10
! 290: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 291: INFO = -12
! 292: END IF
! 293: END IF
! 294: END IF
! 295: *
! 296: IF( INFO.NE.0 ) THEN
! 297: CALL XERBLA( 'ZPPSVX', -INFO )
! 298: RETURN
! 299: END IF
! 300: *
! 301: IF( EQUIL ) THEN
! 302: *
! 303: * Compute row and column scalings to equilibrate the matrix A.
! 304: *
! 305: CALL ZPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU )
! 306: IF( INFEQU.EQ.0 ) THEN
! 307: *
! 308: * Equilibrate the matrix.
! 309: *
! 310: CALL ZLAQHP( UPLO, N, AP, S, SCOND, AMAX, EQUED )
! 311: RCEQU = LSAME( EQUED, 'Y' )
! 312: END IF
! 313: END IF
! 314: *
! 315: * Scale the right-hand side.
! 316: *
! 317: IF( RCEQU ) THEN
! 318: DO 30 J = 1, NRHS
! 319: DO 20 I = 1, N
! 320: B( I, J ) = S( I )*B( I, J )
! 321: 20 CONTINUE
! 322: 30 CONTINUE
! 323: END IF
! 324: *
! 325: IF( NOFACT .OR. EQUIL ) THEN
! 326: *
! 327: * Compute the Cholesky factorization A = U'*U or A = L*L'.
! 328: *
! 329: CALL ZCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
! 330: CALL ZPPTRF( UPLO, N, AFP, INFO )
! 331: *
! 332: * Return if INFO is non-zero.
! 333: *
! 334: IF( INFO.GT.0 )THEN
! 335: RCOND = ZERO
! 336: RETURN
! 337: END IF
! 338: END IF
! 339: *
! 340: * Compute the norm of the matrix A.
! 341: *
! 342: ANORM = ZLANHP( 'I', UPLO, N, AP, RWORK )
! 343: *
! 344: * Compute the reciprocal of the condition number of A.
! 345: *
! 346: CALL ZPPCON( UPLO, N, AFP, ANORM, RCOND, WORK, RWORK, INFO )
! 347: *
! 348: * Compute the solution matrix X.
! 349: *
! 350: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
! 351: CALL ZPPTRS( UPLO, N, NRHS, AFP, X, LDX, INFO )
! 352: *
! 353: * Use iterative refinement to improve the computed solution and
! 354: * compute error bounds and backward error estimates for it.
! 355: *
! 356: CALL ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR,
! 357: $ WORK, RWORK, INFO )
! 358: *
! 359: * Transform the solution matrix X to a solution of the original
! 360: * system.
! 361: *
! 362: IF( RCEQU ) THEN
! 363: DO 50 J = 1, NRHS
! 364: DO 40 I = 1, N
! 365: X( I, J ) = S( I )*X( I, J )
! 366: 40 CONTINUE
! 367: 50 CONTINUE
! 368: DO 60 J = 1, NRHS
! 369: FERR( J ) = FERR( J ) / SCOND
! 370: 60 CONTINUE
! 371: END IF
! 372: *
! 373: * Set INFO = N+1 if the matrix is singular to working precision.
! 374: *
! 375: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
! 376: $ INFO = N + 1
! 377: *
! 378: RETURN
! 379: *
! 380: * End of ZPPSVX
! 381: *
! 382: END
CVSweb interface <joel.bertrand@systella.fr>
![[BACK]](/cvsweb/icons/back.gif){kind=icon}
Return to zppsvx.f CVS log ![[TXT]](/cvsweb/icons/text.gif){kind=icon}
![[DIR]](/cvsweb/icons/dir.gif){kind=icon}
Up to [local] / rpl / lapack / lapack