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140 lines
4.5 KiB
Fortran
140 lines
4.5 KiB
Fortran
*DECK TRED3
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SUBROUTINE TRED3 (N, NV, A, D, E, E2)
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C***BEGIN PROLOGUE TRED3
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C***PURPOSE Reduce a real symmetric matrix stored in packed form to
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C symmetric tridiagonal matrix using orthogonal
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C transformations.
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C***LIBRARY SLATEC (EISPACK)
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C***CATEGORY D4C1B1
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C***TYPE SINGLE PRECISION (TRED3-S)
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C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
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C***AUTHOR Smith, B. T., et al.
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C***DESCRIPTION
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C
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C This subroutine is a translation of the ALGOL procedure TRED3,
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C NUM. MATH. 11, 181-195(1968) by Martin, Reinsch, and Wilkinson.
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C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
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C
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C This subroutine reduces a REAL SYMMETRIC matrix, stored as
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C a one-dimensional array, to a symmetric tridiagonal matrix
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C using orthogonal similarity transformations.
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C
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C On Input
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C
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C N is the order of the matrix A. N is an INTEGER variable.
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C
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C NV is an INTEGER variable set equal to the dimension of the
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C array A as specified in the calling program. NV must not
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C be less than N*(N+1)/2.
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C
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C A contains the lower triangle, stored row-wise, of the real
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C symmetric packed matrix. A is a one-dimensional REAL
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C array, dimensioned A(NV).
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C
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C On Output
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C
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C A contains information about the orthogonal transformations
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C used in the reduction in its first N*(N+1)/2 positions.
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C
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C D contains the diagonal elements of the symmetric tridiagonal
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C matrix. D is a one-dimensional REAL array, dimensioned D(N).
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C
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C E contains the subdiagonal elements of the symmetric
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C tridiagonal matrix in its last N-1 positions. E(1) is set
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C to zero. E is a one-dimensional REAL array, dimensioned
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C E(N).
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C
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C E2 contains the squares of the corresponding elements of E.
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C E2 may coincide with E if the squares are not needed.
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C E2 is a one-dimensional REAL array, dimensioned E2(N).
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C
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C Questions and comments should be directed to B. S. Garbow,
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C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
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C ------------------------------------------------------------------
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C
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C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
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C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
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C system Routines - EISPACK Guide, Springer-Verlag,
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C 1976.
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C***ROUTINES CALLED (NONE)
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C***REVISION HISTORY (YYMMDD)
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C 760101 DATE WRITTEN
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C 890831 Modified array declarations. (WRB)
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C 890831 REVISION DATE from Version 3.2
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C 891214 Prologue converted to Version 4.0 format. (BAB)
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C 920501 Reformatted the REFERENCES section. (WRB)
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C***END PROLOGUE TRED3
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C
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INTEGER I,J,K,L,N,II,IZ,JK,NV
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REAL A(*),D(*),E(*),E2(*)
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REAL F,G,H,HH,SCALE
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C
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C .......... FOR I=N STEP -1 UNTIL 1 DO -- ..........
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C***FIRST EXECUTABLE STATEMENT TRED3
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DO 300 II = 1, N
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I = N + 1 - II
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L = I - 1
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IZ = (I * L) / 2
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H = 0.0E0
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SCALE = 0.0E0
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IF (L .LT. 1) GO TO 130
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C .......... SCALE ROW (ALGOL TOL THEN NOT NEEDED) ..........
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DO 120 K = 1, L
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IZ = IZ + 1
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D(K) = A(IZ)
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SCALE = SCALE + ABS(D(K))
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120 CONTINUE
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C
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IF (SCALE .NE. 0.0E0) GO TO 140
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130 E(I) = 0.0E0
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E2(I) = 0.0E0
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GO TO 290
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C
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140 DO 150 K = 1, L
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D(K) = D(K) / SCALE
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H = H + D(K) * D(K)
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150 CONTINUE
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C
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E2(I) = SCALE * SCALE * H
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F = D(L)
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G = -SIGN(SQRT(H),F)
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E(I) = SCALE * G
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H = H - F * G
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D(L) = F - G
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A(IZ) = SCALE * D(L)
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IF (L .EQ. 1) GO TO 290
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F = 0.0E0
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C
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DO 240 J = 1, L
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G = 0.0E0
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JK = (J * (J-1)) / 2
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C .......... FORM ELEMENT OF A*U ..........
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DO 180 K = 1, L
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JK = JK + 1
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IF (K .GT. J) JK = JK + K - 2
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G = G + A(JK) * D(K)
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180 CONTINUE
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C .......... FORM ELEMENT OF P ..........
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E(J) = G / H
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F = F + E(J) * D(J)
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240 CONTINUE
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C
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HH = F / (H + H)
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JK = 0
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C .......... FORM REDUCED A ..........
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DO 260 J = 1, L
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F = D(J)
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G = E(J) - HH * F
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E(J) = G
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C
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DO 260 K = 1, J
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JK = JK + 1
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A(JK) = A(JK) - F * E(K) - G * D(K)
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260 CONTINUE
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C
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290 D(I) = A(IZ+1)
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A(IZ+1) = SCALE * SQRT(H)
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300 CONTINUE
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C
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RETURN
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END
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