Annotation of rpl/lapack/lapack/zpbtrf.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZPBTRF( UPLO, N, KD, AB, LDAB, INFO )
! 2: *
! 3: * -- LAPACK routine (version 3.2) --
! 4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 6: * November 2006
! 7: *
! 8: * .. Scalar Arguments ..
! 9: CHARACTER UPLO
! 10: INTEGER INFO, KD, LDAB, N
! 11: * ..
! 12: * .. Array Arguments ..
! 13: COMPLEX*16 AB( LDAB, * )
! 14: * ..
! 15: *
! 16: * Purpose
! 17: * =======
! 18: *
! 19: * ZPBTRF computes the Cholesky factorization of a complex Hermitian
! 20: * positive definite band matrix A.
! 21: *
! 22: * The factorization has the form
! 23: * A = U**H * U, if UPLO = 'U', or
! 24: * A = L * L**H, if UPLO = 'L',
! 25: * where U is an upper triangular matrix and L is lower triangular.
! 26: *
! 27: * Arguments
! 28: * =========
! 29: *
! 30: * UPLO (input) CHARACTER*1
! 31: * = 'U': Upper triangle of A is stored;
! 32: * = 'L': Lower triangle of A is stored.
! 33: *
! 34: * N (input) INTEGER
! 35: * The order of the matrix A. N >= 0.
! 36: *
! 37: * KD (input) INTEGER
! 38: * The number of superdiagonals of the matrix A if UPLO = 'U',
! 39: * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
! 40: *
! 41: * AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
! 42: * On entry, the upper or lower triangle of the Hermitian band
! 43: * matrix A, stored in the first KD+1 rows of the array. The
! 44: * j-th column of A is stored in the j-th column of the array AB
! 45: * as follows:
! 46: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
! 47: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
! 48: *
! 49: * On exit, if INFO = 0, the triangular factor U or L from the
! 50: * Cholesky factorization A = U**H*U or A = L*L**H of the band
! 51: * matrix A, in the same storage format as A.
! 52: *
! 53: * LDAB (input) INTEGER
! 54: * The leading dimension of the array AB. LDAB >= KD+1.
! 55: *
! 56: * INFO (output) INTEGER
! 57: * = 0: successful exit
! 58: * < 0: if INFO = -i, the i-th argument had an illegal value
! 59: * > 0: if INFO = i, the leading minor of order i is not
! 60: * positive definite, and the factorization could not be
! 61: * completed.
! 62: *
! 63: * Further Details
! 64: * ===============
! 65: *
! 66: * The band storage scheme is illustrated by the following example, when
! 67: * N = 6, KD = 2, and UPLO = 'U':
! 68: *
! 69: * On entry: On exit:
! 70: *
! 71: * * * a13 a24 a35 a46 * * u13 u24 u35 u46
! 72: * * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
! 73: * a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
! 74: *
! 75: * Similarly, if UPLO = 'L' the format of A is as follows:
! 76: *
! 77: * On entry: On exit:
! 78: *
! 79: * a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
! 80: * a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
! 81: * a31 a42 a53 a64 * * l31 l42 l53 l64 * *
! 82: *
! 83: * Array elements marked * are not used by the routine.
! 84: *
! 85: * Contributed by
! 86: * Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
! 87: *
! 88: * =====================================================================
! 89: *
! 90: * .. Parameters ..
! 91: DOUBLE PRECISION ONE, ZERO
! 92: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
! 93: COMPLEX*16 CONE
! 94: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
! 95: INTEGER NBMAX, LDWORK
! 96: PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 )
! 97: * ..
! 98: * .. Local Scalars ..
! 99: INTEGER I, I2, I3, IB, II, J, JJ, NB
! 100: * ..
! 101: * .. Local Arrays ..
! 102: COMPLEX*16 WORK( LDWORK, NBMAX )
! 103: * ..
! 104: * .. External Functions ..
! 105: LOGICAL LSAME
! 106: INTEGER ILAENV
! 107: EXTERNAL LSAME, ILAENV
! 108: * ..
! 109: * .. External Subroutines ..
! 110: EXTERNAL XERBLA, ZGEMM, ZHERK, ZPBTF2, ZPOTF2, ZTRSM
! 111: * ..
! 112: * .. Intrinsic Functions ..
! 113: INTRINSIC MIN
! 114: * ..
! 115: * .. Executable Statements ..
! 116: *
! 117: * Test the input parameters.
! 118: *
! 119: INFO = 0
! 120: IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND.
! 121: $ ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN
! 122: INFO = -1
! 123: ELSE IF( N.LT.0 ) THEN
! 124: INFO = -2
! 125: ELSE IF( KD.LT.0 ) THEN
! 126: INFO = -3
! 127: ELSE IF( LDAB.LT.KD+1 ) THEN
! 128: INFO = -5
! 129: END IF
! 130: IF( INFO.NE.0 ) THEN
! 131: CALL XERBLA( 'ZPBTRF', -INFO )
! 132: RETURN
! 133: END IF
! 134: *
! 135: * Quick return if possible
! 136: *
! 137: IF( N.EQ.0 )
! 138: $ RETURN
! 139: *
! 140: * Determine the block size for this environment
! 141: *
! 142: NB = ILAENV( 1, 'ZPBTRF', UPLO, N, KD, -1, -1 )
! 143: *
! 144: * The block size must not exceed the semi-bandwidth KD, and must not
! 145: * exceed the limit set by the size of the local array WORK.
! 146: *
! 147: NB = MIN( NB, NBMAX )
! 148: *
! 149: IF( NB.LE.1 .OR. NB.GT.KD ) THEN
! 150: *
! 151: * Use unblocked code
! 152: *
! 153: CALL ZPBTF2( UPLO, N, KD, AB, LDAB, INFO )
! 154: ELSE
! 155: *
! 156: * Use blocked code
! 157: *
! 158: IF( LSAME( UPLO, 'U' ) ) THEN
! 159: *
! 160: * Compute the Cholesky factorization of a Hermitian band
! 161: * matrix, given the upper triangle of the matrix in band
! 162: * storage.
! 163: *
! 164: * Zero the upper triangle of the work array.
! 165: *
! 166: DO 20 J = 1, NB
! 167: DO 10 I = 1, J - 1
! 168: WORK( I, J ) = ZERO
! 169: 10 CONTINUE
! 170: 20 CONTINUE
! 171: *
! 172: * Process the band matrix one diagonal block at a time.
! 173: *
! 174: DO 70 I = 1, N, NB
! 175: IB = MIN( NB, N-I+1 )
! 176: *
! 177: * Factorize the diagonal block
! 178: *
! 179: CALL ZPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II )
! 180: IF( II.NE.0 ) THEN
! 181: INFO = I + II - 1
! 182: GO TO 150
! 183: END IF
! 184: IF( I+IB.LE.N ) THEN
! 185: *
! 186: * Update the relevant part of the trailing submatrix.
! 187: * If A11 denotes the diagonal block which has just been
! 188: * factorized, then we need to update the remaining
! 189: * blocks in the diagram:
! 190: *
! 191: * A11 A12 A13
! 192: * A22 A23
! 193: * A33
! 194: *
! 195: * The numbers of rows and columns in the partitioning
! 196: * are IB, I2, I3 respectively. The blocks A12, A22 and
! 197: * A23 are empty if IB = KD. The upper triangle of A13
! 198: * lies outside the band.
! 199: *
! 200: I2 = MIN( KD-IB, N-I-IB+1 )
! 201: I3 = MIN( IB, N-I-KD+1 )
! 202: *
! 203: IF( I2.GT.0 ) THEN
! 204: *
! 205: * Update A12
! 206: *
! 207: CALL ZTRSM( 'Left', 'Upper', 'Conjugate transpose',
! 208: $ 'Non-unit', IB, I2, CONE,
! 209: $ AB( KD+1, I ), LDAB-1,
! 210: $ AB( KD+1-IB, I+IB ), LDAB-1 )
! 211: *
! 212: * Update A22
! 213: *
! 214: CALL ZHERK( 'Upper', 'Conjugate transpose', I2, IB,
! 215: $ -ONE, AB( KD+1-IB, I+IB ), LDAB-1, ONE,
! 216: $ AB( KD+1, I+IB ), LDAB-1 )
! 217: END IF
! 218: *
! 219: IF( I3.GT.0 ) THEN
! 220: *
! 221: * Copy the lower triangle of A13 into the work array.
! 222: *
! 223: DO 40 JJ = 1, I3
! 224: DO 30 II = JJ, IB
! 225: WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 )
! 226: 30 CONTINUE
! 227: 40 CONTINUE
! 228: *
! 229: * Update A13 (in the work array).
! 230: *
! 231: CALL ZTRSM( 'Left', 'Upper', 'Conjugate transpose',
! 232: $ 'Non-unit', IB, I3, CONE,
! 233: $ AB( KD+1, I ), LDAB-1, WORK, LDWORK )
! 234: *
! 235: * Update A23
! 236: *
! 237: IF( I2.GT.0 )
! 238: $ CALL ZGEMM( 'Conjugate transpose',
! 239: $ 'No transpose', I2, I3, IB, -CONE,
! 240: $ AB( KD+1-IB, I+IB ), LDAB-1, WORK,
! 241: $ LDWORK, CONE, AB( 1+IB, I+KD ),
! 242: $ LDAB-1 )
! 243: *
! 244: * Update A33
! 245: *
! 246: CALL ZHERK( 'Upper', 'Conjugate transpose', I3, IB,
! 247: $ -ONE, WORK, LDWORK, ONE,
! 248: $ AB( KD+1, I+KD ), LDAB-1 )
! 249: *
! 250: * Copy the lower triangle of A13 back into place.
! 251: *
! 252: DO 60 JJ = 1, I3
! 253: DO 50 II = JJ, IB
! 254: AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ )
! 255: 50 CONTINUE
! 256: 60 CONTINUE
! 257: END IF
! 258: END IF
! 259: 70 CONTINUE
! 260: ELSE
! 261: *
! 262: * Compute the Cholesky factorization of a Hermitian band
! 263: * matrix, given the lower triangle of the matrix in band
! 264: * storage.
! 265: *
! 266: * Zero the lower triangle of the work array.
! 267: *
! 268: DO 90 J = 1, NB
! 269: DO 80 I = J + 1, NB
! 270: WORK( I, J ) = ZERO
! 271: 80 CONTINUE
! 272: 90 CONTINUE
! 273: *
! 274: * Process the band matrix one diagonal block at a time.
! 275: *
! 276: DO 140 I = 1, N, NB
! 277: IB = MIN( NB, N-I+1 )
! 278: *
! 279: * Factorize the diagonal block
! 280: *
! 281: CALL ZPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II )
! 282: IF( II.NE.0 ) THEN
! 283: INFO = I + II - 1
! 284: GO TO 150
! 285: END IF
! 286: IF( I+IB.LE.N ) THEN
! 287: *
! 288: * Update the relevant part of the trailing submatrix.
! 289: * If A11 denotes the diagonal block which has just been
! 290: * factorized, then we need to update the remaining
! 291: * blocks in the diagram:
! 292: *
! 293: * A11
! 294: * A21 A22
! 295: * A31 A32 A33
! 296: *
! 297: * The numbers of rows and columns in the partitioning
! 298: * are IB, I2, I3 respectively. The blocks A21, A22 and
! 299: * A32 are empty if IB = KD. The lower triangle of A31
! 300: * lies outside the band.
! 301: *
! 302: I2 = MIN( KD-IB, N-I-IB+1 )
! 303: I3 = MIN( IB, N-I-KD+1 )
! 304: *
! 305: IF( I2.GT.0 ) THEN
! 306: *
! 307: * Update A21
! 308: *
! 309: CALL ZTRSM( 'Right', 'Lower',
! 310: $ 'Conjugate transpose', 'Non-unit', I2,
! 311: $ IB, CONE, AB( 1, I ), LDAB-1,
! 312: $ AB( 1+IB, I ), LDAB-1 )
! 313: *
! 314: * Update A22
! 315: *
! 316: CALL ZHERK( 'Lower', 'No transpose', I2, IB, -ONE,
! 317: $ AB( 1+IB, I ), LDAB-1, ONE,
! 318: $ AB( 1, I+IB ), LDAB-1 )
! 319: END IF
! 320: *
! 321: IF( I3.GT.0 ) THEN
! 322: *
! 323: * Copy the upper triangle of A31 into the work array.
! 324: *
! 325: DO 110 JJ = 1, IB
! 326: DO 100 II = 1, MIN( JJ, I3 )
! 327: WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 )
! 328: 100 CONTINUE
! 329: 110 CONTINUE
! 330: *
! 331: * Update A31 (in the work array).
! 332: *
! 333: CALL ZTRSM( 'Right', 'Lower',
! 334: $ 'Conjugate transpose', 'Non-unit', I3,
! 335: $ IB, CONE, AB( 1, I ), LDAB-1, WORK,
! 336: $ LDWORK )
! 337: *
! 338: * Update A32
! 339: *
! 340: IF( I2.GT.0 )
! 341: $ CALL ZGEMM( 'No transpose',
! 342: $ 'Conjugate transpose', I3, I2, IB,
! 343: $ -CONE, WORK, LDWORK, AB( 1+IB, I ),
! 344: $ LDAB-1, CONE, AB( 1+KD-IB, I+IB ),
! 345: $ LDAB-1 )
! 346: *
! 347: * Update A33
! 348: *
! 349: CALL ZHERK( 'Lower', 'No transpose', I3, IB, -ONE,
! 350: $ WORK, LDWORK, ONE, AB( 1, I+KD ),
! 351: $ LDAB-1 )
! 352: *
! 353: * Copy the upper triangle of A31 back into place.
! 354: *
! 355: DO 130 JJ = 1, IB
! 356: DO 120 II = 1, MIN( JJ, I3 )
! 357: AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ )
! 358: 120 CONTINUE
! 359: 130 CONTINUE
! 360: END IF
! 361: END IF
! 362: 140 CONTINUE
! 363: END IF
! 364: END IF
! 365: RETURN
! 366: *
! 367: 150 CONTINUE
! 368: RETURN
! 369: *
! 370: * End of ZPBTRF
! 371: *
! 372: END
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