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