### Table 4: Model diagnostics

"... In PAGE 6: ... Given the results in Table 3 it seems sensible to conclude that there exist a unique cointegrating vector at the zero frequency and two vectors at the annual and biannual frequencies. Since the various test statistics rely on uncorrelated and normal distributed residuals Table4 reports univariate and multivariate misspeci cation tests for the VAR. The results indicate no severe problems with autocorrelation and normality.... ..."

### Table 1. Inversion formula Inversion algorithm Algorithm for solving

1995

"... In PAGE 2: ...Boros-Kailath-Olshevskyalgorithm [BKO] Three-term Verde{Star formula [VS] Calvetti{Reichel Heinig-Hoppe-Rost Vandermonde and algorithm [CR] algorithm [HHR] and matrices Gohberg{Olshevsky Higham algorithm formula [GO2] [Hig1], [Hig2] All the algorithms in Table1 are called fast algorithms, because their complexity of O(n2) op- erations compares favorably with the O(n3) operations of general purpose algorithms like Gaussian elimination. Now inversion and fast solution of a linear system are two classical applications of the concept of displacement structure, see [KS2].... In PAGE 2: ...1), but also certain natural generalizations thereof, which we shall call polynomial Vandermonde-like matrices. We shall see that the concept allows us to nicely unify and extend the results in Table1 . The main reason is that the displacement structure is essentially preserved under inversion, see [KKM] and [KS2].... In PAGE 4: ... However, a natural question is how these three new classes of matrices relate to each other? We show that no matter which of the three displacement operators is chosen, all three de nitions lead in fact to the same class of matrices, which we shall call the class of polynomial Vandermonde-like matrices. Furthermore we show that all the results in the bottom line of Table1 can be carried over to this wider class of matrices. In particular, we derive (a) two inversion formulas; (b) a structured implementation of Gaussian elimination with partial pivoting, and (c) an inversion algorithm for polynomial Vandermonde-like matrices.... ..."

Cited by 14

### Table 4. Inversion formula Inversion algorithm Algorithm for solving

1995

"... In PAGE 4: ...All the algorithms in Table4 have complexity O( n2) operations, where is the displacement rank of a matrix. 0.... ..."

Cited by 14

### Table 1. Comparison of Calculated Asymptotic Plate Velocity

2003

"... In PAGE 6: ... The velocity ramping was not captured in either Zapotec calculation, with the asymptotic velocity noticeably over-predicted with the coarser CTH mesh and only a slight over-prediction for the finer mesh calculation. Numeric values for the calculated asymptotic velocities are provided in Table1 . Figure 6.... In PAGE 7: ...effective shell thickness. The results are also provided in Table1 . As the scale factor is increased, the expanded shell covers more CTH cells along the top direction.... ..."

Cited by 1

### Table 3. Comparison of Eiy, amp;n calculated by Rayleigh-Schrtidinger perturbation theory and by the asymptotic formula

in i

### Table 1: Efficiency of Lagrange procedure for BU BPBIand BK.

"... In PAGE 4: ... In the absence of an analytical expression for the number of iter- ations required to arrive at a solution, we demonstrate the efficiency of the proposed procedure empirically. In Table1 , we list the total number D4 BU B4CFB5 of unique TAM partitions for a total TAM width of CF and for BU TAMs. The value of D4 BU B4CFB5 is calculated using the expression D4 BU B4CFB5BP CF BUA0BD BUAXB4BUA0BDB5AX [8].... In PAGE 4: ... Note that this expression is accurate only for larger values of CF; hence we present results only for CF AL BGBG. In Table1 , we compare the efficiency of the La- grange optimization algorithm with that of the Partition evaluate algorithm proposed to solve Problem C8 C6C8BTCF in [8]. The efficiency AH is calculated as the ratio of the number of TAM partitions evalu- ated by the Lagrange optimization procedure to the total number of unique partitions.... ..."

### Table 1: Efficiency of Lagrange procedure for BU BPBIand BK.

"... In PAGE 4: ... In the absence of an analytical expression for the number of iter- ations required to arrive at a solution, we demonstrate the efficiency of the proposed procedure empirically. In Table1 , we list the total number D4 BU B4CFB5 of unique TAM partitions for a total TAM width of CF and for BU TAMs. The value of D4 BU B4CFB5 is calculated using the expression D4 BU B4CFB5BP CF BUA0BD BUAXB4BUA0BDB5AX [8].... In PAGE 4: ... Note that this expression is accurate only for larger values of CF; hence we present results only for CF AL BGBG. In Table1 , we compare the efficiency of the La- grange optimization algorithm with that of the Partition evaluate algorithm proposed to solve Problem C8 C6C8BTCF in [8]. The efficiency AH is calculated as the ratio of the number of TAM partitions evalu- ated by the Lagrange optimization procedure to the total number of unique partitions.... ..."

### Table 2: Efficiency of the Lagrange optimization procedure for BU BPBIand BK.

2004

"... In PAGE 18: ... In the absence of an analytical expression for the number of iterations required to arrive at a solution, we demonstrate the efficiency of the proposed procedure empirically. In Table2 , we list the total number D4 BU B4CF B5 of unique TAM partitions for a total TAM width of CF and for BU TAMs. The value of D4 BU B4CF B5 is calculated using the expression D4 BU B4CF B5 BP CF BUA0BD BUAXB4BUA0BDB5AX [12].... ..."

Cited by 4

### Table 3 Formulas for calculating WO and WI.

in A method by

"... In PAGE 6: ... As the backtrace goes from a gate g to a gate i driving one of its inputs, WO and WI of gate i ( WO, and WI,) are adjusted depending on the logical function of gate g. The weights for gate i resulting from the path from gate g are given in Table3 for the primitive logic functions of gate g, where Ri is the value calculated from Equation (1). The new value of WOi is the larger of WO calculated from Table 1 and the previous value of WO,.... ..."

### Table 1b: Same as Table 1a but for Lagrange FEM of 3rd order.

"... In PAGE 21: ... The mesh size has been kept xed with boundaries at rmin = 0 fm and rmax = 10 fm. In Table1 a neutron single particle energies which have been calculated with linear nite elements are shown for the initial Woods-Saxon potential. For increasing number of mesh points (see left column), the number of unchanged decimal places reaches 8 at 200 mesh points.... In PAGE 21: ... For increasing number of mesh points (see left column), the number of unchanged decimal places reaches 8 at 200 mesh points. A comparison with the last row of Table1 b shows that for linear elements the last digit (decimal place 10) has not stabilized at the extremely large mesh point number 600. Table 1b displays results that have been calculated with nite elements of 3rd order.... In PAGE 21: ... A comparison with the last row of Table 1b shows that for linear elements the last digit (decimal place 10) has not stabilized at the extremely large mesh point number 600. Table1 b displays results that have been calculated with nite elements of 3rd order. Between 109 and 121 mesh points (36-40 elements), the results have stabilized in all 10 digits.... In PAGE 23: ...Table 1c: Same as Table1 b but for 4th order Lagrange FEM.... In PAGE 24: ...1860841159 45.2876261622 Table 1d: Same as Table1 d but for 5th order Lagrange FEM. 6.... In PAGE 24: ...1860841159 45.2876261622 Table 1e: Same as Table1 d but for 6th order Lagrange FEM.... In PAGE 25: ...1860841159 45.2876261622 Table 1f: Same as Table1 e but for 7th order Lagrange FEM. Table 1d displays eigenvalues which have been calculated with nite elements of 5th order.... In PAGE 25: ...2876261622 Table 1f: Same as Table 1e but for 7th order Lagrange FEM. Table1 d displays eigenvalues which have been calculated with nite elements of 5th order. At 76 mesh points 12 digits have stabilized for all 6 eigenvalues.... In PAGE 25: ... At 76 mesh points 12 digits have stabilized for all 6 eigenvalues. At 41 mesh points the precision is already as good as the precision in Table1 a at 600 mesh points. A comparison of the eigenvalues in Table 1d with results of a 6th order FEM calculation in Table 1e shows that a further increase of the order leads to a rather weak reduction of the number of required mesh points.... In PAGE 25: ... At 41 mesh points the precision is already as good as the precision in Table 1a at 600 mesh points. A comparison of the eigenvalues in Table 1d with results of a 6th order FEM calculation in Table1 e shows that a further increase of the order leads to a rather weak reduction of the number of required mesh points. At least 73 mesh points are necessary in 6th order for a precision of 12 digits.... In PAGE 25: ... At least 73 mesh points are necessary in 6th order for a precision of 12 digits. As shown in Table1 f, the reduction of the number of mesh points is even weaker when the order is increased from 6th order to 7th order. In the subsequent Tables 2a to 2f, results of corresponding calculations with B-spline nite elements are shown.... In PAGE 25: ... In Table 2a, neutron single particle eigenvalues are listed which have been calculated with the new B-spline FEM code. A comparison of the numbers with those listed in Table1 a shows that they are identical for equal mesh point numbers. For increasing order of the B-splines, the number of required mesh points to obtain a certain precision reduces very similarly to the trend observed in the Tables 1a to 1f.... In PAGE 25: ... A comparison of the Tables 2a to 2f with the corresponding Tables 1a to 1f shows that roughly half the number of mesh points is required in a B-spline FEM in order to achieve the precision of a corresponding calculation with Lagrangian nite elements. In Table1 b, full precision is achieved at 60 mesh points while 121 mesh points were necessary in Table 1b. In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table 2c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits.... In PAGE 25: ... A comparison of the Tables 2a to 2f with the corresponding Tables 1a to 1f shows that roughly half the number of mesh points is required in a B-spline FEM in order to achieve the precision of a corresponding calculation with Lagrangian nite elements. In Table 1b, full precision is achieved at 60 mesh points while 121 mesh points were necessary in Table1 b. In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table 2c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits.... In PAGE 25: ... In Table 1b, full precision is achieved at 60 mesh points while 121 mesh points were necessary in Table 1b. In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table 2c whereas 145 mesh points are necessary with Lagrange elements ( Table1 c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d).... In PAGE 25: ... In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table 2c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points ( Table1 d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f).... In PAGE 26: ...18608424 45.28762620 Table 2a: Same as Table1 a but for B-spline FEM. 3.... In PAGE 26: ...18608412 45.28762616 Table 2b: Same as Table1... In PAGE 27: ...1860841159 45.2876261622 Table 2c: Same as Table1 c but for B-spline FEM. 5.... In PAGE 27: ...1860841159 45.2876261622 Table 2d: Same as Table1... In PAGE 28: ...1860841159 45.2876261622 Table 2e: Same as Table1 e but for B-spline FEM. 7.... In PAGE 28: ...1860841159 45.2876261622 Table 2f: Same as Table1 f but for B-spline FEM. To complete this study, I repeated the calculations for a large number of mesh points with both methods from 1st order to 8th order nite elements.... ..."