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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) 2: * 3: * -- LAPACK auxiliary 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 LDA, LDW, N, NB 11: * .. 12: * .. Array Arguments .. 13: DOUBLE PRECISION E( * ) 14: COMPLEX*16 A( LDA, * ), TAU( * ), W( LDW, * ) 15: * .. 16: * 17: * Purpose 18: * ======= 19: * 20: * ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to 21: * Hermitian tridiagonal form by a unitary similarity 22: * transformation Q' * A * Q, and returns the matrices V and W which are 23: * needed to apply the transformation to the unreduced part of A. 24: * 25: * If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a 26: * matrix, of which the upper triangle is supplied; 27: * if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a 28: * matrix, of which the lower triangle is supplied. 29: * 30: * This is an auxiliary routine called by ZHETRD. 31: * 32: * Arguments 33: * ========= 34: * 35: * UPLO (input) CHARACTER*1 36: * Specifies whether the upper or lower triangular part of the 37: * Hermitian matrix A is stored: 38: * = 'U': Upper triangular 39: * = 'L': Lower triangular 40: * 41: * N (input) INTEGER 42: * The order of the matrix A. 43: * 44: * NB (input) INTEGER 45: * The number of rows and columns to be reduced. 46: * 47: * A (input/output) COMPLEX*16 array, dimension (LDA,N) 48: * On entry, the Hermitian matrix A. If UPLO = 'U', the leading 49: * n-by-n upper triangular part of A contains the upper 50: * triangular part of the matrix A, and the strictly lower 51: * triangular part of A is not referenced. If UPLO = 'L', the 52: * leading n-by-n lower triangular part of A contains the lower 53: * triangular part of the matrix A, and the strictly upper 54: * triangular part of A is not referenced. 55: * On exit: 56: * if UPLO = 'U', the last NB columns have been reduced to 57: * tridiagonal form, with the diagonal elements overwriting 58: * the diagonal elements of A; the elements above the diagonal 59: * with the array TAU, represent the unitary matrix Q as a 60: * product of elementary reflectors; 61: * if UPLO = 'L', the first NB columns have been reduced to 62: * tridiagonal form, with the diagonal elements overwriting 63: * the diagonal elements of A; the elements below the diagonal 64: * with the array TAU, represent the unitary matrix Q as a 65: * product of elementary reflectors. 66: * See Further Details. 67: * 68: * LDA (input) INTEGER 69: * The leading dimension of the array A. LDA >= max(1,N). 70: * 71: * E (output) DOUBLE PRECISION array, dimension (N-1) 72: * If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal 73: * elements of the last NB columns of the reduced matrix; 74: * if UPLO = 'L', E(1:nb) contains the subdiagonal elements of 75: * the first NB columns of the reduced matrix. 76: * 77: * TAU (output) COMPLEX*16 array, dimension (N-1) 78: * The scalar factors of the elementary reflectors, stored in 79: * TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. 80: * See Further Details. 81: * 82: * W (output) COMPLEX*16 array, dimension (LDW,NB) 83: * The n-by-nb matrix W required to update the unreduced part 84: * of A. 85: * 86: * LDW (input) INTEGER 87: * The leading dimension of the array W. LDW >= max(1,N). 88: * 89: * Further Details 90: * =============== 91: * 92: * If UPLO = 'U', the matrix Q is represented as a product of elementary 93: * reflectors 94: * 95: * Q = H(n) H(n-1) . . . H(n-nb+1). 96: * 97: * Each H(i) has the form 98: * 99: * H(i) = I - tau * v * v' 100: * 101: * where tau is a complex scalar, and v is a complex vector with 102: * v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), 103: * and tau in TAU(i-1). 104: * 105: * If UPLO = 'L', the matrix Q is represented as a product of elementary 106: * reflectors 107: * 108: * Q = H(1) H(2) . . . H(nb). 109: * 110: * Each H(i) has the form 111: * 112: * H(i) = I - tau * v * v' 113: * 114: * where tau is a complex scalar, and v is a complex vector with 115: * v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), 116: * and tau in TAU(i). 117: * 118: * The elements of the vectors v together form the n-by-nb matrix V 119: * which is needed, with W, to apply the transformation to the unreduced 120: * part of the matrix, using a Hermitian rank-2k update of the form: 121: * A := A - V*W' - W*V'. 122: * 123: * The contents of A on exit are illustrated by the following examples 124: * with n = 5 and nb = 2: 125: * 126: * if UPLO = 'U': if UPLO = 'L': 127: * 128: * ( a a a v4 v5 ) ( d ) 129: * ( a a v4 v5 ) ( 1 d ) 130: * ( a 1 v5 ) ( v1 1 a ) 131: * ( d 1 ) ( v1 v2 a a ) 132: * ( d ) ( v1 v2 a a a ) 133: * 134: * where d denotes a diagonal element of the reduced matrix, a denotes 135: * an element of the original matrix that is unchanged, and vi denotes 136: * an element of the vector defining H(i). 137: * 138: * ===================================================================== 139: * 140: * .. Parameters .. 141: COMPLEX*16 ZERO, ONE, HALF 142: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), 143: $ ONE = ( 1.0D+0, 0.0D+0 ), 144: $ HALF = ( 0.5D+0, 0.0D+0 ) ) 145: * .. 146: * .. Local Scalars .. 147: INTEGER I, IW 148: COMPLEX*16 ALPHA 149: * .. 150: * .. External Subroutines .. 151: EXTERNAL ZAXPY, ZGEMV, ZHEMV, ZLACGV, ZLARFG, ZSCAL 152: * .. 153: * .. External Functions .. 154: LOGICAL LSAME 155: COMPLEX*16 ZDOTC 156: EXTERNAL LSAME, ZDOTC 157: * .. 158: * .. Intrinsic Functions .. 159: INTRINSIC DBLE, MIN 160: * .. 161: * .. Executable Statements .. 162: * 163: * Quick return if possible 164: * 165: IF( N.LE.0 ) 166: $ RETURN 167: * 168: IF( LSAME( UPLO, 'U' ) ) THEN 169: * 170: * Reduce last NB columns of upper triangle 171: * 172: DO 10 I = N, N - NB + 1, -1 173: IW = I - N + NB 174: IF( I.LT.N ) THEN 175: * 176: * Update A(1:i,i) 177: * 178: A( I, I ) = DBLE( A( I, I ) ) 179: CALL ZLACGV( N-I, W( I, IW+1 ), LDW ) 180: CALL ZGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ), 181: $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 ) 182: CALL ZLACGV( N-I, W( I, IW+1 ), LDW ) 183: CALL ZLACGV( N-I, A( I, I+1 ), LDA ) 184: CALL ZGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ), 185: $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 ) 186: CALL ZLACGV( N-I, A( I, I+1 ), LDA ) 187: A( I, I ) = DBLE( A( I, I ) ) 188: END IF 189: IF( I.GT.1 ) THEN 190: * 191: * Generate elementary reflector H(i) to annihilate 192: * A(1:i-2,i) 193: * 194: ALPHA = A( I-1, I ) 195: CALL ZLARFG( I-1, ALPHA, A( 1, I ), 1, TAU( I-1 ) ) 196: E( I-1 ) = ALPHA 197: A( I-1, I ) = ONE 198: * 199: * Compute W(1:i-1,i) 200: * 201: CALL ZHEMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1, 202: $ ZERO, W( 1, IW ), 1 ) 203: IF( I.LT.N ) THEN 204: CALL ZGEMV( 'Conjugate transpose', I-1, N-I, ONE, 205: $ W( 1, IW+1 ), LDW, A( 1, I ), 1, ZERO, 206: $ W( I+1, IW ), 1 ) 207: CALL ZGEMV( 'No transpose', I-1, N-I, -ONE, 208: $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE, 209: $ W( 1, IW ), 1 ) 210: CALL ZGEMV( 'Conjugate transpose', I-1, N-I, ONE, 211: $ A( 1, I+1 ), LDA, A( 1, I ), 1, ZERO, 212: $ W( I+1, IW ), 1 ) 213: CALL ZGEMV( 'No transpose', I-1, N-I, -ONE, 214: $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE, 215: $ W( 1, IW ), 1 ) 216: END IF 217: CALL ZSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 ) 218: ALPHA = -HALF*TAU( I-1 )*ZDOTC( I-1, W( 1, IW ), 1, 219: $ A( 1, I ), 1 ) 220: CALL ZAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 ) 221: END IF 222: * 223: 10 CONTINUE 224: ELSE 225: * 226: * Reduce first NB columns of lower triangle 227: * 228: DO 20 I = 1, NB 229: * 230: * Update A(i:n,i) 231: * 232: A( I, I ) = DBLE( A( I, I ) ) 233: CALL ZLACGV( I-1, W( I, 1 ), LDW ) 234: CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ), 235: $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 ) 236: CALL ZLACGV( I-1, W( I, 1 ), LDW ) 237: CALL ZLACGV( I-1, A( I, 1 ), LDA ) 238: CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ), 239: $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 ) 240: CALL ZLACGV( I-1, A( I, 1 ), LDA ) 241: A( I, I ) = DBLE( A( I, I ) ) 242: IF( I.LT.N ) THEN 243: * 244: * Generate elementary reflector H(i) to annihilate 245: * A(i+2:n,i) 246: * 247: ALPHA = A( I+1, I ) 248: CALL ZLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1, 249: $ TAU( I ) ) 250: E( I ) = ALPHA 251: A( I+1, I ) = ONE 252: * 253: * Compute W(i+1:n,i) 254: * 255: CALL ZHEMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA, 256: $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 ) 257: CALL ZGEMV( 'Conjugate transpose', N-I, I-1, ONE, 258: $ W( I+1, 1 ), LDW, A( I+1, I ), 1, ZERO, 259: $ W( 1, I ), 1 ) 260: CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ), 261: $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) 262: CALL ZGEMV( 'Conjugate transpose', N-I, I-1, ONE, 263: $ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO, 264: $ W( 1, I ), 1 ) 265: CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ), 266: $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) 267: CALL ZSCAL( N-I, TAU( I ), W( I+1, I ), 1 ) 268: ALPHA = -HALF*TAU( I )*ZDOTC( N-I, W( I+1, I ), 1, 269: $ A( I+1, I ), 1 ) 270: CALL ZAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 ) 271: END IF 272: * 273: 20 CONTINUE 274: END IF 275: * 276: RETURN 277: * 278: * End of ZLATRD 279: * 280: END