Annotation of rpl/lapack/lapack/dpftri.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DPFTRI( TRANSR, UPLO, N, A, INFO )
! 2: *
! 3: * -- LAPACK routine (version 3.2.2) --
! 4: *
! 5: * -- Contributed by Fred Gustavson of the IBM Watson Research Center --
! 6: * -- June 2010 --
! 7: *
! 8: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 9: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 10: *
! 11: * .. Scalar Arguments ..
! 12: CHARACTER TRANSR, UPLO
! 13: INTEGER INFO, N
! 14: * .. Array Arguments ..
! 15: DOUBLE PRECISION A( 0: * )
! 16: * ..
! 17: *
! 18: * Purpose
! 19: * =======
! 20: *
! 21: * DPFTRI computes the inverse of a (real) symmetric positive definite
! 22: * matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
! 23: * computed by DPFTRF.
! 24: *
! 25: * Arguments
! 26: * =========
! 27: *
! 28: * TRANSR (input) CHARACTER
! 29: * = 'N': The Normal TRANSR of RFP A is stored;
! 30: * = 'T': The Transpose TRANSR of RFP A is stored.
! 31: *
! 32: * UPLO (input) CHARACTER
! 33: * = 'U': Upper triangle of A is stored;
! 34: * = 'L': Lower triangle of A is stored.
! 35: *
! 36: * N (input) INTEGER
! 37: * The order of the matrix A. N >= 0.
! 38: *
! 39: * A (input/output) DOUBLE PRECISION array, dimension ( N*(N+1)/2 )
! 40: * On entry, the symmetric matrix A in RFP format. RFP format is
! 41: * described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
! 42: * then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
! 43: * (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
! 44: * the transpose of RFP A as defined when
! 45: * TRANSR = 'N'. The contents of RFP A are defined by UPLO as
! 46: * follows: If UPLO = 'U' the RFP A contains the nt elements of
! 47: * upper packed A. If UPLO = 'L' the RFP A contains the elements
! 48: * of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
! 49: * 'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
! 50: * is odd. See the Note below for more details.
! 51: *
! 52: * On exit, the symmetric inverse of the original matrix, in the
! 53: * same storage format.
! 54: *
! 55: * INFO (output) INTEGER
! 56: * = 0: successful exit
! 57: * < 0: if INFO = -i, the i-th argument had an illegal value
! 58: * > 0: if INFO = i, the (i,i) element of the factor U or L is
! 59: * zero, and the inverse could not be computed.
! 60: *
! 61: * Further Details
! 62: * ===============
! 63: *
! 64: * We first consider Rectangular Full Packed (RFP) Format when N is
! 65: * even. We give an example where N = 6.
! 66: *
! 67: * AP is Upper AP is Lower
! 68: *
! 69: * 00 01 02 03 04 05 00
! 70: * 11 12 13 14 15 10 11
! 71: * 22 23 24 25 20 21 22
! 72: * 33 34 35 30 31 32 33
! 73: * 44 45 40 41 42 43 44
! 74: * 55 50 51 52 53 54 55
! 75: *
! 76: *
! 77: * Let TRANSR = 'N'. RFP holds AP as follows:
! 78: * For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
! 79: * three columns of AP upper. The lower triangle A(4:6,0:2) consists of
! 80: * the transpose of the first three columns of AP upper.
! 81: * For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
! 82: * three columns of AP lower. The upper triangle A(0:2,0:2) consists of
! 83: * the transpose of the last three columns of AP lower.
! 84: * This covers the case N even and TRANSR = 'N'.
! 85: *
! 86: * RFP A RFP A
! 87: *
! 88: * 03 04 05 33 43 53
! 89: * 13 14 15 00 44 54
! 90: * 23 24 25 10 11 55
! 91: * 33 34 35 20 21 22
! 92: * 00 44 45 30 31 32
! 93: * 01 11 55 40 41 42
! 94: * 02 12 22 50 51 52
! 95: *
! 96: * Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
! 97: * transpose of RFP A above. One therefore gets:
! 98: *
! 99: *
! 100: * RFP A RFP A
! 101: *
! 102: * 03 13 23 33 00 01 02 33 00 10 20 30 40 50
! 103: * 04 14 24 34 44 11 12 43 44 11 21 31 41 51
! 104: * 05 15 25 35 45 55 22 53 54 55 22 32 42 52
! 105: *
! 106: *
! 107: * We then consider Rectangular Full Packed (RFP) Format when N is
! 108: * odd. We give an example where N = 5.
! 109: *
! 110: * AP is Upper AP is Lower
! 111: *
! 112: * 00 01 02 03 04 00
! 113: * 11 12 13 14 10 11
! 114: * 22 23 24 20 21 22
! 115: * 33 34 30 31 32 33
! 116: * 44 40 41 42 43 44
! 117: *
! 118: *
! 119: * Let TRANSR = 'N'. RFP holds AP as follows:
! 120: * For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
! 121: * three columns of AP upper. The lower triangle A(3:4,0:1) consists of
! 122: * the transpose of the first two columns of AP upper.
! 123: * For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
! 124: * three columns of AP lower. The upper triangle A(0:1,1:2) consists of
! 125: * the transpose of the last two columns of AP lower.
! 126: * This covers the case N odd and TRANSR = 'N'.
! 127: *
! 128: * RFP A RFP A
! 129: *
! 130: * 02 03 04 00 33 43
! 131: * 12 13 14 10 11 44
! 132: * 22 23 24 20 21 22
! 133: * 00 33 34 30 31 32
! 134: * 01 11 44 40 41 42
! 135: *
! 136: * Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
! 137: * transpose of RFP A above. One therefore gets:
! 138: *
! 139: * RFP A RFP A
! 140: *
! 141: * 02 12 22 00 01 00 10 20 30 40 50
! 142: * 03 13 23 33 11 33 11 21 31 41 51
! 143: * 04 14 24 34 44 43 44 22 32 42 52
! 144: *
! 145: * =====================================================================
! 146: *
! 147: * .. Parameters ..
! 148: DOUBLE PRECISION ONE
! 149: PARAMETER ( ONE = 1.0D+0 )
! 150: * ..
! 151: * .. Local Scalars ..
! 152: LOGICAL LOWER, NISODD, NORMALTRANSR
! 153: INTEGER N1, N2, K
! 154: * ..
! 155: * .. External Functions ..
! 156: LOGICAL LSAME
! 157: EXTERNAL LSAME
! 158: * ..
! 159: * .. External Subroutines ..
! 160: EXTERNAL XERBLA, DTFTRI, DLAUUM, DTRMM, DSYRK
! 161: * ..
! 162: * .. Intrinsic Functions ..
! 163: INTRINSIC MOD
! 164: * ..
! 165: * .. Executable Statements ..
! 166: *
! 167: * Test the input parameters.
! 168: *
! 169: INFO = 0
! 170: NORMALTRANSR = LSAME( TRANSR, 'N' )
! 171: LOWER = LSAME( UPLO, 'L' )
! 172: IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
! 173: INFO = -1
! 174: ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
! 175: INFO = -2
! 176: ELSE IF( N.LT.0 ) THEN
! 177: INFO = -3
! 178: END IF
! 179: IF( INFO.NE.0 ) THEN
! 180: CALL XERBLA( 'DPFTRI', -INFO )
! 181: RETURN
! 182: END IF
! 183: *
! 184: * Quick return if possible
! 185: *
! 186: IF( N.EQ.0 )
! 187: + RETURN
! 188: *
! 189: * Invert the triangular Cholesky factor U or L.
! 190: *
! 191: CALL DTFTRI( TRANSR, UPLO, 'N', N, A, INFO )
! 192: IF( INFO.GT.0 )
! 193: + RETURN
! 194: *
! 195: * If N is odd, set NISODD = .TRUE.
! 196: * If N is even, set K = N/2 and NISODD = .FALSE.
! 197: *
! 198: IF( MOD( N, 2 ).EQ.0 ) THEN
! 199: K = N / 2
! 200: NISODD = .FALSE.
! 201: ELSE
! 202: NISODD = .TRUE.
! 203: END IF
! 204: *
! 205: * Set N1 and N2 depending on LOWER
! 206: *
! 207: IF( LOWER ) THEN
! 208: N2 = N / 2
! 209: N1 = N - N2
! 210: ELSE
! 211: N1 = N / 2
! 212: N2 = N - N1
! 213: END IF
! 214: *
! 215: * Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
! 216: * inv(L)^C*inv(L). There are eight cases.
! 217: *
! 218: IF( NISODD ) THEN
! 219: *
! 220: * N is odd
! 221: *
! 222: IF( NORMALTRANSR ) THEN
! 223: *
! 224: * N is odd and TRANSR = 'N'
! 225: *
! 226: IF( LOWER ) THEN
! 227: *
! 228: * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
! 229: * T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
! 230: * T1 -> a(0), T2 -> a(n), S -> a(N1)
! 231: *
! 232: CALL DLAUUM( 'L', N1, A( 0 ), N, INFO )
! 233: CALL DSYRK( 'L', 'T', N1, N2, ONE, A( N1 ), N, ONE,
! 234: + A( 0 ), N )
! 235: CALL DTRMM( 'L', 'U', 'N', 'N', N2, N1, ONE, A( N ), N,
! 236: + A( N1 ), N )
! 237: CALL DLAUUM( 'U', N2, A( N ), N, INFO )
! 238: *
! 239: ELSE
! 240: *
! 241: * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
! 242: * T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
! 243: * T1 -> a(N2), T2 -> a(N1), S -> a(0)
! 244: *
! 245: CALL DLAUUM( 'L', N1, A( N2 ), N, INFO )
! 246: CALL DSYRK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE,
! 247: + A( N2 ), N )
! 248: CALL DTRMM( 'R', 'U', 'T', 'N', N1, N2, ONE, A( N1 ), N,
! 249: + A( 0 ), N )
! 250: CALL DLAUUM( 'U', N2, A( N1 ), N, INFO )
! 251: *
! 252: END IF
! 253: *
! 254: ELSE
! 255: *
! 256: * N is odd and TRANSR = 'T'
! 257: *
! 258: IF( LOWER ) THEN
! 259: *
! 260: * SRPA for LOWER, TRANSPOSE, and N is odd
! 261: * T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
! 262: *
! 263: CALL DLAUUM( 'U', N1, A( 0 ), N1, INFO )
! 264: CALL DSYRK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE,
! 265: + A( 0 ), N1 )
! 266: CALL DTRMM( 'R', 'L', 'N', 'N', N1, N2, ONE, A( 1 ), N1,
! 267: + A( N1*N1 ), N1 )
! 268: CALL DLAUUM( 'L', N2, A( 1 ), N1, INFO )
! 269: *
! 270: ELSE
! 271: *
! 272: * SRPA for UPPER, TRANSPOSE, and N is odd
! 273: * T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
! 274: *
! 275: CALL DLAUUM( 'U', N1, A( N2*N2 ), N2, INFO )
! 276: CALL DSYRK( 'U', 'T', N1, N2, ONE, A( 0 ), N2, ONE,
! 277: + A( N2*N2 ), N2 )
! 278: CALL DTRMM( 'L', 'L', 'T', 'N', N2, N1, ONE, A( N1*N2 ),
! 279: + N2, A( 0 ), N2 )
! 280: CALL DLAUUM( 'L', N2, A( N1*N2 ), N2, INFO )
! 281: *
! 282: END IF
! 283: *
! 284: END IF
! 285: *
! 286: ELSE
! 287: *
! 288: * N is even
! 289: *
! 290: IF( NORMALTRANSR ) THEN
! 291: *
! 292: * N is even and TRANSR = 'N'
! 293: *
! 294: IF( LOWER ) THEN
! 295: *
! 296: * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
! 297: * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
! 298: * T1 -> a(1), T2 -> a(0), S -> a(k+1)
! 299: *
! 300: CALL DLAUUM( 'L', K, A( 1 ), N+1, INFO )
! 301: CALL DSYRK( 'L', 'T', K, K, ONE, A( K+1 ), N+1, ONE,
! 302: + A( 1 ), N+1 )
! 303: CALL DTRMM( 'L', 'U', 'N', 'N', K, K, ONE, A( 0 ), N+1,
! 304: + A( K+1 ), N+1 )
! 305: CALL DLAUUM( 'U', K, A( 0 ), N+1, INFO )
! 306: *
! 307: ELSE
! 308: *
! 309: * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
! 310: * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
! 311: * T1 -> a(k+1), T2 -> a(k), S -> a(0)
! 312: *
! 313: CALL DLAUUM( 'L', K, A( K+1 ), N+1, INFO )
! 314: CALL DSYRK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE,
! 315: + A( K+1 ), N+1 )
! 316: CALL DTRMM( 'R', 'U', 'T', 'N', K, K, ONE, A( K ), N+1,
! 317: + A( 0 ), N+1 )
! 318: CALL DLAUUM( 'U', K, A( K ), N+1, INFO )
! 319: *
! 320: END IF
! 321: *
! 322: ELSE
! 323: *
! 324: * N is even and TRANSR = 'T'
! 325: *
! 326: IF( LOWER ) THEN
! 327: *
! 328: * SRPA for LOWER, TRANSPOSE, and N is even (see paper)
! 329: * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
! 330: * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
! 331: *
! 332: CALL DLAUUM( 'U', K, A( K ), K, INFO )
! 333: CALL DSYRK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE,
! 334: + A( K ), K )
! 335: CALL DTRMM( 'R', 'L', 'N', 'N', K, K, ONE, A( 0 ), K,
! 336: + A( K*( K+1 ) ), K )
! 337: CALL DLAUUM( 'L', K, A( 0 ), K, INFO )
! 338: *
! 339: ELSE
! 340: *
! 341: * SRPA for UPPER, TRANSPOSE, and N is even (see paper)
! 342: * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0),
! 343: * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
! 344: *
! 345: CALL DLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO )
! 346: CALL DSYRK( 'U', 'T', K, K, ONE, A( 0 ), K, ONE,
! 347: + A( K*( K+1 ) ), K )
! 348: CALL DTRMM( 'L', 'L', 'T', 'N', K, K, ONE, A( K*K ), K,
! 349: + A( 0 ), K )
! 350: CALL DLAUUM( 'L', K, A( K*K ), K, INFO )
! 351: *
! 352: END IF
! 353: *
! 354: END IF
! 355: *
! 356: END IF
! 357: *
! 358: RETURN
! 359: *
! 360: * End of DPFTRI
! 361: *
! 362: END
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