Annotation of rpl/lapack/lapack/dtprfs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
! 2: $ FERR, BERR, WORK, IWORK, INFO )
! 3: *
! 4: * -- LAPACK 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: * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
! 10: *
! 11: * .. Scalar Arguments ..
! 12: CHARACTER DIAG, TRANS, UPLO
! 13: INTEGER INFO, LDB, LDX, N, NRHS
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IWORK( * )
! 17: DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
! 18: $ WORK( * ), X( LDX, * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * DTPRFS provides error bounds and backward error estimates for the
! 25: * solution to a system of linear equations with a triangular packed
! 26: * coefficient matrix.
! 27: *
! 28: * The solution matrix X must be computed by DTPTRS or some other
! 29: * means before entering this routine. DTPRFS does not do iterative
! 30: * refinement because doing so cannot improve the backward error.
! 31: *
! 32: * Arguments
! 33: * =========
! 34: *
! 35: * UPLO (input) CHARACTER*1
! 36: * = 'U': A is upper triangular;
! 37: * = 'L': A is lower triangular.
! 38: *
! 39: * TRANS (input) CHARACTER*1
! 40: * Specifies the form of the system of equations:
! 41: * = 'N': A * X = B (No transpose)
! 42: * = 'T': A**T * X = B (Transpose)
! 43: * = 'C': A**H * X = B (Conjugate transpose = Transpose)
! 44: *
! 45: * DIAG (input) CHARACTER*1
! 46: * = 'N': A is non-unit triangular;
! 47: * = 'U': A is unit triangular.
! 48: *
! 49: * N (input) INTEGER
! 50: * The order of the matrix A. N >= 0.
! 51: *
! 52: * NRHS (input) INTEGER
! 53: * The number of right hand sides, i.e., the number of columns
! 54: * of the matrices B and X. NRHS >= 0.
! 55: *
! 56: * AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
! 57: * The upper or lower triangular matrix A, packed columnwise in
! 58: * a linear array. The j-th column of A is stored in the array
! 59: * AP as follows:
! 60: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 61: * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
! 62: * If DIAG = 'U', the diagonal elements of A are not referenced
! 63: * and are assumed to be 1.
! 64: *
! 65: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 66: * The right hand side matrix B.
! 67: *
! 68: * LDB (input) INTEGER
! 69: * The leading dimension of the array B. LDB >= max(1,N).
! 70: *
! 71: * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 72: * The solution matrix X.
! 73: *
! 74: * LDX (input) INTEGER
! 75: * The leading dimension of the array X. LDX >= max(1,N).
! 76: *
! 77: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 78: * The estimated forward error bound for each solution vector
! 79: * X(j) (the j-th column of the solution matrix X).
! 80: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 81: * is an estimated upper bound for the magnitude of the largest
! 82: * element in (X(j) - XTRUE) divided by the magnitude of the
! 83: * largest element in X(j). The estimate is as reliable as
! 84: * the estimate for RCOND, and is almost always a slight
! 85: * overestimate of the true error.
! 86: *
! 87: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 88: * The componentwise relative backward error of each solution
! 89: * vector X(j) (i.e., the smallest relative change in
! 90: * any element of A or B that makes X(j) an exact solution).
! 91: *
! 92: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
! 93: *
! 94: * IWORK (workspace) INTEGER array, dimension (N)
! 95: *
! 96: * INFO (output) INTEGER
! 97: * = 0: successful exit
! 98: * < 0: if INFO = -i, the i-th argument had an illegal value
! 99: *
! 100: * =====================================================================
! 101: *
! 102: * .. Parameters ..
! 103: DOUBLE PRECISION ZERO
! 104: PARAMETER ( ZERO = 0.0D+0 )
! 105: DOUBLE PRECISION ONE
! 106: PARAMETER ( ONE = 1.0D+0 )
! 107: * ..
! 108: * .. Local Scalars ..
! 109: LOGICAL NOTRAN, NOUNIT, UPPER
! 110: CHARACTER TRANST
! 111: INTEGER I, J, K, KASE, KC, NZ
! 112: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
! 113: * ..
! 114: * .. Local Arrays ..
! 115: INTEGER ISAVE( 3 )
! 116: * ..
! 117: * .. External Subroutines ..
! 118: EXTERNAL DAXPY, DCOPY, DLACN2, DTPMV, DTPSV, XERBLA
! 119: * ..
! 120: * .. Intrinsic Functions ..
! 121: INTRINSIC ABS, MAX
! 122: * ..
! 123: * .. External Functions ..
! 124: LOGICAL LSAME
! 125: DOUBLE PRECISION DLAMCH
! 126: EXTERNAL LSAME, DLAMCH
! 127: * ..
! 128: * .. Executable Statements ..
! 129: *
! 130: * Test the input parameters.
! 131: *
! 132: INFO = 0
! 133: UPPER = LSAME( UPLO, 'U' )
! 134: NOTRAN = LSAME( TRANS, 'N' )
! 135: NOUNIT = LSAME( DIAG, 'N' )
! 136: *
! 137: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 138: INFO = -1
! 139: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
! 140: $ LSAME( TRANS, 'C' ) ) THEN
! 141: INFO = -2
! 142: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
! 143: INFO = -3
! 144: ELSE IF( N.LT.0 ) THEN
! 145: INFO = -4
! 146: ELSE IF( NRHS.LT.0 ) THEN
! 147: INFO = -5
! 148: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 149: INFO = -8
! 150: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 151: INFO = -10
! 152: END IF
! 153: IF( INFO.NE.0 ) THEN
! 154: CALL XERBLA( 'DTPRFS', -INFO )
! 155: RETURN
! 156: END IF
! 157: *
! 158: * Quick return if possible
! 159: *
! 160: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
! 161: DO 10 J = 1, NRHS
! 162: FERR( J ) = ZERO
! 163: BERR( J ) = ZERO
! 164: 10 CONTINUE
! 165: RETURN
! 166: END IF
! 167: *
! 168: IF( NOTRAN ) THEN
! 169: TRANST = 'T'
! 170: ELSE
! 171: TRANST = 'N'
! 172: END IF
! 173: *
! 174: * NZ = maximum number of nonzero elements in each row of A, plus 1
! 175: *
! 176: NZ = N + 1
! 177: EPS = DLAMCH( 'Epsilon' )
! 178: SAFMIN = DLAMCH( 'Safe minimum' )
! 179: SAFE1 = NZ*SAFMIN
! 180: SAFE2 = SAFE1 / EPS
! 181: *
! 182: * Do for each right hand side
! 183: *
! 184: DO 250 J = 1, NRHS
! 185: *
! 186: * Compute residual R = B - op(A) * X,
! 187: * where op(A) = A or A', depending on TRANS.
! 188: *
! 189: CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
! 190: CALL DTPMV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
! 191: CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
! 192: *
! 193: * Compute componentwise relative backward error from formula
! 194: *
! 195: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
! 196: *
! 197: * where abs(Z) is the componentwise absolute value of the matrix
! 198: * or vector Z. If the i-th component of the denominator is less
! 199: * than SAFE2, then SAFE1 is added to the i-th components of the
! 200: * numerator and denominator before dividing.
! 201: *
! 202: DO 20 I = 1, N
! 203: WORK( I ) = ABS( B( I, J ) )
! 204: 20 CONTINUE
! 205: *
! 206: IF( NOTRAN ) THEN
! 207: *
! 208: * Compute abs(A)*abs(X) + abs(B).
! 209: *
! 210: IF( UPPER ) THEN
! 211: KC = 1
! 212: IF( NOUNIT ) THEN
! 213: DO 40 K = 1, N
! 214: XK = ABS( X( K, J ) )
! 215: DO 30 I = 1, K
! 216: WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
! 217: 30 CONTINUE
! 218: KC = KC + K
! 219: 40 CONTINUE
! 220: ELSE
! 221: DO 60 K = 1, N
! 222: XK = ABS( X( K, J ) )
! 223: DO 50 I = 1, K - 1
! 224: WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
! 225: 50 CONTINUE
! 226: WORK( K ) = WORK( K ) + XK
! 227: KC = KC + K
! 228: 60 CONTINUE
! 229: END IF
! 230: ELSE
! 231: KC = 1
! 232: IF( NOUNIT ) THEN
! 233: DO 80 K = 1, N
! 234: XK = ABS( X( K, J ) )
! 235: DO 70 I = K, N
! 236: WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
! 237: 70 CONTINUE
! 238: KC = KC + N - K + 1
! 239: 80 CONTINUE
! 240: ELSE
! 241: DO 100 K = 1, N
! 242: XK = ABS( X( K, J ) )
! 243: DO 90 I = K + 1, N
! 244: WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
! 245: 90 CONTINUE
! 246: WORK( K ) = WORK( K ) + XK
! 247: KC = KC + N - K + 1
! 248: 100 CONTINUE
! 249: END IF
! 250: END IF
! 251: ELSE
! 252: *
! 253: * Compute abs(A')*abs(X) + abs(B).
! 254: *
! 255: IF( UPPER ) THEN
! 256: KC = 1
! 257: IF( NOUNIT ) THEN
! 258: DO 120 K = 1, N
! 259: S = ZERO
! 260: DO 110 I = 1, K
! 261: S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
! 262: 110 CONTINUE
! 263: WORK( K ) = WORK( K ) + S
! 264: KC = KC + K
! 265: 120 CONTINUE
! 266: ELSE
! 267: DO 140 K = 1, N
! 268: S = ABS( X( K, J ) )
! 269: DO 130 I = 1, K - 1
! 270: S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
! 271: 130 CONTINUE
! 272: WORK( K ) = WORK( K ) + S
! 273: KC = KC + K
! 274: 140 CONTINUE
! 275: END IF
! 276: ELSE
! 277: KC = 1
! 278: IF( NOUNIT ) THEN
! 279: DO 160 K = 1, N
! 280: S = ZERO
! 281: DO 150 I = K, N
! 282: S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
! 283: 150 CONTINUE
! 284: WORK( K ) = WORK( K ) + S
! 285: KC = KC + N - K + 1
! 286: 160 CONTINUE
! 287: ELSE
! 288: DO 180 K = 1, N
! 289: S = ABS( X( K, J ) )
! 290: DO 170 I = K + 1, N
! 291: S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
! 292: 170 CONTINUE
! 293: WORK( K ) = WORK( K ) + S
! 294: KC = KC + N - K + 1
! 295: 180 CONTINUE
! 296: END IF
! 297: END IF
! 298: END IF
! 299: S = ZERO
! 300: DO 190 I = 1, N
! 301: IF( WORK( I ).GT.SAFE2 ) THEN
! 302: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
! 303: ELSE
! 304: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
! 305: $ ( WORK( I )+SAFE1 ) )
! 306: END IF
! 307: 190 CONTINUE
! 308: BERR( J ) = S
! 309: *
! 310: * Bound error from formula
! 311: *
! 312: * norm(X - XTRUE) / norm(X) .le. FERR =
! 313: * norm( abs(inv(op(A)))*
! 314: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
! 315: *
! 316: * where
! 317: * norm(Z) is the magnitude of the largest component of Z
! 318: * inv(op(A)) is the inverse of op(A)
! 319: * abs(Z) is the componentwise absolute value of the matrix or
! 320: * vector Z
! 321: * NZ is the maximum number of nonzeros in any row of A, plus 1
! 322: * EPS is machine epsilon
! 323: *
! 324: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
! 325: * is incremented by SAFE1 if the i-th component of
! 326: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
! 327: *
! 328: * Use DLACN2 to estimate the infinity-norm of the matrix
! 329: * inv(op(A)) * diag(W),
! 330: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
! 331: *
! 332: DO 200 I = 1, N
! 333: IF( WORK( I ).GT.SAFE2 ) THEN
! 334: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
! 335: ELSE
! 336: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
! 337: END IF
! 338: 200 CONTINUE
! 339: *
! 340: KASE = 0
! 341: 210 CONTINUE
! 342: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
! 343: $ KASE, ISAVE )
! 344: IF( KASE.NE.0 ) THEN
! 345: IF( KASE.EQ.1 ) THEN
! 346: *
! 347: * Multiply by diag(W)*inv(op(A)').
! 348: *
! 349: CALL DTPSV( UPLO, TRANST, DIAG, N, AP, WORK( N+1 ), 1 )
! 350: DO 220 I = 1, N
! 351: WORK( N+I ) = WORK( I )*WORK( N+I )
! 352: 220 CONTINUE
! 353: ELSE
! 354: *
! 355: * Multiply by inv(op(A))*diag(W).
! 356: *
! 357: DO 230 I = 1, N
! 358: WORK( N+I ) = WORK( I )*WORK( N+I )
! 359: 230 CONTINUE
! 360: CALL DTPSV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
! 361: END IF
! 362: GO TO 210
! 363: END IF
! 364: *
! 365: * Normalize error.
! 366: *
! 367: LSTRES = ZERO
! 368: DO 240 I = 1, N
! 369: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
! 370: 240 CONTINUE
! 371: IF( LSTRES.NE.ZERO )
! 372: $ FERR( J ) = FERR( J ) / LSTRES
! 373: *
! 374: 250 CONTINUE
! 375: *
! 376: RETURN
! 377: *
! 378: * End of DTPRFS
! 379: *
! 380: END
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