Refinable C1 spline elements for irregular quad layout - UF CISE ...
Refinable C 1 spline elements for irregular quad layout Thien Nguyen
¨ Peters Jorg
University of Florida
NSF CCF-0728797, NIH R01-LM011300
T. Nguyen, J. Peters (UF)
GMP 2016
1 / 20
Outline
1
Refinable, smooth, CAD compatible spline space incl. irregularities
2
Algorithm
3
Applications
T. Nguyen, J. Peters (UF)
GMP 2016
2 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Outline
1
Refinable, smooth, CAD compatible spline space incl. irregularities
2
Algorithm
3
Applications
T. Nguyen, J. Peters (UF)
GMP 2016
3 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible
I
multi-sided blends, irregularities
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible I I
multi-sided blends, irregularities subdivision surface: , nested space
/ infinite rings; / industrial design infrastructure; / integration rules;
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible I I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules; Gk spline complex: , industrial design infrastructure
/ refinement book keeping (non-local); / or: not nested : problem for free-form surfaces! T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible I I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules;
Gk spline complex: , industrial design infrastructure / refinement book keeping (non-local); not nested Challenge: combine, for multi-sided configurations, splines with simple nested refinability.
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible I I
I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules; Gk spline complex: , industrial design infrastructure / refinement book keeping (non-local); not nested
singularly parameterized surface , nested space, , industrial design infrastructure (Peters 91, Neamtu 94) (Reif 97) , proves C 1 surface (projection)
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible
I I
I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules;
Gk spline complex: , industrial design infrastructure / refinement book keeping (non-local); not nested
singularly parameterized surface , nested space, industrial design infrastructure (Reif 97) , proves C 1 surface (projection)
/ fewer d.o.f. near irregularity than in regular regions
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible
I I
I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules;
Gk spline complex: , industrial design infrastructure / refinement book keeping (non-local); not nested
singularly parameterized surface , nested space, industrial design infrastructure (Reif 97) , proves C 1 surface (projection)
/ fewer d.o.f. near irregularity than in regular regions / d.o.f. can not be symmetrically distributed as proper control points.
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible
I I
I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules;
Gk spline complex: , industrial design infrastructure / refinement book keeping (non-local); not nested
singularly parameterized surface , nested space, industrial design infrastructure (Reif 97) , proves C 1 surface (projection)
/ fewer d.o.f. near irregularity than in regular regions / d.o.f. can not be symmetrically distributed as proper control points. / Surface shape is poor. T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
2 × 2 split construction I
2 × 2 split yields uniform d.o.f.: , regardless of vertex valences, each quad has 4 d.o.f.!
T. Nguyen, J. Peters (UF)
GMP 2016
5 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
2 × 2 split construction
I
I
2 × 2 split yields uniform d.o.f.: , regardless of vertex valences, each quad has 4 d.o.f.!
C 1 bi-3 basis functions , naturally compatible with bi-cubic PHT refinement
T. Nguyen, J. Peters (UF)
GMP 2016
5 / 20
Algorithm
Outline
1
Refinable, smooth, CAD compatible spline space incl. irregularities
2
Algorithm
3
Applications
T. Nguyen, J. Peters (UF)
GMP 2016
6 / 20
Algorithm
Algorithm Input
Input: B-spline-like control points cij` I
Recall: regular double-knot bi-3 B-spline coefficients are co-located with 1b 2b01 ´ ”inner” Bezier coefficients: c11 → 14 00 2b10 4b11 T. Nguyen, J. Peters (UF)
GMP 2016
7 / 20
Algorithm
Algorithm Output
k ,11 ´ Output: Bezier points bαβ obtained by projection P
T. Nguyen, J. Peters (UF)
GMP 2016
8 / 20
Algorithm
Algorithm: PSc I
Conversion to BB form: copy aijk := cijk for i, j ∈ {1, 2}
k make C 1 except a00 :=
T. Nguyen, J. Peters (UF)
Pn
k=1
k c11 /n.
GMP 2016
9 / 20
Algorithm
Algorithm: PSc I
Conversion to BB form: copy aijk := cijk for i, j ∈ {1, 2}
k make C 1 except a00 := I
Pn
k=1
k c11 /n.
Subdivide hk ,ij := Sak , i, j ∈ {1, 2}. When i + j > 2 then bk,ij := hk,ij .
T. Nguyen, J. Peters (UF)
GMP 2016
9 / 20
Algorithm
Algorithm: PSc I
Conversion to BB form: copy aijk := cijk for i, j ∈ {1, 2} Pn k k /n. make C 1 except a00 := k=1 c11
I
Subdivide hk ,ij := Sak , i, j ∈ {1, 2}. When i + j > 2 then bk,ij := hk,ij .
I
k ,11 Subpatches hαβ := hαβ :
b11 h11 b21 := P h21 b12 h12 For all edges make C 1 : k+1 k k b10 := (b11 + b11 /2).
T. Nguyen, J. Peters (UF)
GMP 2016
9 / 20
Algorithm
Properties of the splines I
singular parameterization at irregularities.
T. Nguyen, J. Peters (UF)
GMP 2016
10 / 20
Algorithm
Properties of the splines I
singular parameterization at irregularities.
I
C 1 continuity (Reif 97: invertible Jacobian → exists regular reparameterization at 0)
T. Nguyen, J. Peters (UF)
GMP 2016
10 / 20
Algorithm
Properties of the splines I
singular parameterization at irregularities.
I
C 1 continuity (Reif 97: invertible Jacobian → exists regular reparameterization at 0)
I
For 1 < p < 4, the Lp norms of the main curvatures are finite
T. Nguyen, J. Peters (UF)
GMP 2016
10 / 20
Algorithm
Properties of the splines I
singular parameterization at irregularities.
I
C 1 continuity (Reif 97: invertible Jacobian → exists regular reparameterization at 0)
I
For 1 < p < 4, the Lp norms of the main curvatures are finite Refinable (nested spaces)
I
S
PSc −−−−→ SPSc yP yP
(1)
S
PSc −−−−→ SPSc (PSPS = SPS since S does not change the projected function)
T. Nguyen, J. Peters (UF)
GMP 2016
10 / 20
Algorithm
Properties of the splines I
singular parameterization at irregularities.
I
C 1 continuity (Reif 97: invertible Jacobian → exists regular reparameterization at 0)
I
For 1 < p < 4, the Lp norms of the main curvatures are finite Refinable (nested spaces)
I
S
PSc −−−−→ SPSc yP yP
(1)
S
PSc −−−−→ SPSc (PSPS = SPS since S does not change the projected function) I
Linear independence of fijk associated cijk (proof via functionals)
T. Nguyen, J. Peters (UF)
GMP 2016
10 / 20
Applications
Outline
1
Refinable, smooth, CAD compatible spline space incl. irregularities
2
Algorithm
3
Applications
T. Nguyen, J. Peters (UF)
GMP 2016
11 / 20
Applications
Applications: free-form surfaces
T. Nguyen, J. Peters (UF)
GMP 2016
12 / 20
Applications
Applications: free-form surfaces
T. Nguyen, J. Peters (UF)
GMP 2016
13 / 20
Applications
Applications: free-form surfaces
T. Nguyen, J. Peters (UF)
GMP 2016
14 / 20
Applications
Applications: Poisson local refinement
Poisson’s equation on the square [0, 6]2
.
Error -0.08:0.08,
T. Nguyen, J. Peters (UF)
-0.02:0.02,
GMP 2016
-0.01:0.01
15 / 20
Applications
Applications: Poisson
Error and approximate convergence rate (a.c.r.): close to 2−4 ` 1 2 3 4 5
||u − uh ||L2 0.0625 0.0098 9.7e-04 7.17e-5 5.29e-06
T. Nguyen, J. Peters (UF)
a.c.r. ||.||L2 6.4 10.1 13.5 13.56
||u − uh ||L∞ 0.0826 0.0194 0.0015 1.3e-4 9.87e-06
GMP 2016
a.c.r. ||.||L∞ 4.3 12.9 11.5 13.1
||u − uh ||H 1 0.4568 0.1448 0.0225 0.0033 4.78e-04
a.c.r.
3 6 6 6
16 / 20
Applications
Applications: Thin shell Scordelis-Lo roof
T. Nguyen, J. Peters (UF)
GMP 2016
17 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
I
Degrees of freedom = as for PHT splines (C 1 bi-3 spline)
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
I
Degrees of freedom = as for PHT splines (C 1 bi-3 spline)
I
, isogeometric analysis
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
I
Degrees of freedom = as for PHT splines (C 1 bi-3 spline)
I I
, isogeometric analysis / shape
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
I
Degrees of freedom = as for PHT splines (C 1 bi-3 spline)
I I
, isogeometric analysis / shape Thank You & Questions?
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
I
Degrees of freedom = as for PHT splines (C 1 bi-3 spline)
I I
, isogeometric analysis / shape Thank You & Questions?
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
T. Nguyen, J. Peters (UF)
GMP 2016
19 / 20
Applications
linear independence
` Nonzero BB coefficients • of the basis function f21 . The coefficient marked ` k k additionally with an × is nonzero only for f21 . It is zero for f12 or f11 , k k = 0, . . . , n − 1 and for f21 , k 6= `.
T. Nguyen, J. Peters (UF)
GMP 2016
20 / 20
¨ Peters Jorg
University of Florida
NSF CCF-0728797, NIH R01-LM011300
T. Nguyen, J. Peters (UF)
GMP 2016
1 / 20
Outline
1
Refinable, smooth, CAD compatible spline space incl. irregularities
2
Algorithm
3
Applications
T. Nguyen, J. Peters (UF)
GMP 2016
2 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Outline
1
Refinable, smooth, CAD compatible spline space incl. irregularities
2
Algorithm
3
Applications
T. Nguyen, J. Peters (UF)
GMP 2016
3 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible
I
multi-sided blends, irregularities
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible I I
multi-sided blends, irregularities subdivision surface: , nested space
/ infinite rings; / industrial design infrastructure; / integration rules;
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible I I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules; Gk spline complex: , industrial design infrastructure
/ refinement book keeping (non-local); / or: not nested : problem for free-form surfaces! T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible I I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules;
Gk spline complex: , industrial design infrastructure / refinement book keeping (non-local); not nested Challenge: combine, for multi-sided configurations, splines with simple nested refinability.
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible I I
I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules; Gk spline complex: , industrial design infrastructure / refinement book keeping (non-local); not nested
singularly parameterized surface , nested space, , industrial design infrastructure (Peters 91, Neamtu 94) (Reif 97) , proves C 1 surface (projection)
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible
I I
I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules;
Gk spline complex: , industrial design infrastructure / refinement book keeping (non-local); not nested
singularly parameterized surface , nested space, industrial design infrastructure (Reif 97) , proves C 1 surface (projection)
/ fewer d.o.f. near irregularity than in regular regions
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible
I I
I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules;
Gk spline complex: , industrial design infrastructure / refinement book keeping (non-local); not nested
singularly parameterized surface , nested space, industrial design infrastructure (Reif 97) , proves C 1 surface (projection)
/ fewer d.o.f. near irregularity than in regular regions / d.o.f. can not be symmetrically distributed as proper control points.
T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
Challenge: refinable, smooth and CAD compatible
I I
I
I
multi-sided blends, irregularities subdivision surface: , nested space / infinite rings; industrial design infrastructure; integration rules;
Gk spline complex: , industrial design infrastructure / refinement book keeping (non-local); not nested
singularly parameterized surface , nested space, industrial design infrastructure (Reif 97) , proves C 1 surface (projection)
/ fewer d.o.f. near irregularity than in regular regions / d.o.f. can not be symmetrically distributed as proper control points. / Surface shape is poor. T. Nguyen, J. Peters (UF)
GMP 2016
4 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
2 × 2 split construction I
2 × 2 split yields uniform d.o.f.: , regardless of vertex valences, each quad has 4 d.o.f.!
T. Nguyen, J. Peters (UF)
GMP 2016
5 / 20
Refinable, smooth, CAD compatible spline space incl. irregularities
2 × 2 split construction
I
I
2 × 2 split yields uniform d.o.f.: , regardless of vertex valences, each quad has 4 d.o.f.!
C 1 bi-3 basis functions , naturally compatible with bi-cubic PHT refinement
T. Nguyen, J. Peters (UF)
GMP 2016
5 / 20
Algorithm
Outline
1
Refinable, smooth, CAD compatible spline space incl. irregularities
2
Algorithm
3
Applications
T. Nguyen, J. Peters (UF)
GMP 2016
6 / 20
Algorithm
Algorithm Input
Input: B-spline-like control points cij` I
Recall: regular double-knot bi-3 B-spline coefficients are co-located with 1b 2b01 ´ ”inner” Bezier coefficients: c11 → 14 00 2b10 4b11 T. Nguyen, J. Peters (UF)
GMP 2016
7 / 20
Algorithm
Algorithm Output
k ,11 ´ Output: Bezier points bαβ obtained by projection P
T. Nguyen, J. Peters (UF)
GMP 2016
8 / 20
Algorithm
Algorithm: PSc I
Conversion to BB form: copy aijk := cijk for i, j ∈ {1, 2}
k make C 1 except a00 :=
T. Nguyen, J. Peters (UF)
Pn
k=1
k c11 /n.
GMP 2016
9 / 20
Algorithm
Algorithm: PSc I
Conversion to BB form: copy aijk := cijk for i, j ∈ {1, 2}
k make C 1 except a00 := I
Pn
k=1
k c11 /n.
Subdivide hk ,ij := Sak , i, j ∈ {1, 2}. When i + j > 2 then bk,ij := hk,ij .
T. Nguyen, J. Peters (UF)
GMP 2016
9 / 20
Algorithm
Algorithm: PSc I
Conversion to BB form: copy aijk := cijk for i, j ∈ {1, 2} Pn k k /n. make C 1 except a00 := k=1 c11
I
Subdivide hk ,ij := Sak , i, j ∈ {1, 2}. When i + j > 2 then bk,ij := hk,ij .
I
k ,11 Subpatches hαβ := hαβ :
b11 h11 b21 := P h21 b12 h12 For all edges make C 1 : k+1 k k b10 := (b11 + b11 /2).
T. Nguyen, J. Peters (UF)
GMP 2016
9 / 20
Algorithm
Properties of the splines I
singular parameterization at irregularities.
T. Nguyen, J. Peters (UF)
GMP 2016
10 / 20
Algorithm
Properties of the splines I
singular parameterization at irregularities.
I
C 1 continuity (Reif 97: invertible Jacobian → exists regular reparameterization at 0)
T. Nguyen, J. Peters (UF)
GMP 2016
10 / 20
Algorithm
Properties of the splines I
singular parameterization at irregularities.
I
C 1 continuity (Reif 97: invertible Jacobian → exists regular reparameterization at 0)
I
For 1 < p < 4, the Lp norms of the main curvatures are finite
T. Nguyen, J. Peters (UF)
GMP 2016
10 / 20
Algorithm
Properties of the splines I
singular parameterization at irregularities.
I
C 1 continuity (Reif 97: invertible Jacobian → exists regular reparameterization at 0)
I
For 1 < p < 4, the Lp norms of the main curvatures are finite Refinable (nested spaces)
I
S
PSc −−−−→ SPSc yP yP
(1)
S
PSc −−−−→ SPSc (PSPS = SPS since S does not change the projected function)
T. Nguyen, J. Peters (UF)
GMP 2016
10 / 20
Algorithm
Properties of the splines I
singular parameterization at irregularities.
I
C 1 continuity (Reif 97: invertible Jacobian → exists regular reparameterization at 0)
I
For 1 < p < 4, the Lp norms of the main curvatures are finite Refinable (nested spaces)
I
S
PSc −−−−→ SPSc yP yP
(1)
S
PSc −−−−→ SPSc (PSPS = SPS since S does not change the projected function) I
Linear independence of fijk associated cijk (proof via functionals)
T. Nguyen, J. Peters (UF)
GMP 2016
10 / 20
Applications
Outline
1
Refinable, smooth, CAD compatible spline space incl. irregularities
2
Algorithm
3
Applications
T. Nguyen, J. Peters (UF)
GMP 2016
11 / 20
Applications
Applications: free-form surfaces
T. Nguyen, J. Peters (UF)
GMP 2016
12 / 20
Applications
Applications: free-form surfaces
T. Nguyen, J. Peters (UF)
GMP 2016
13 / 20
Applications
Applications: free-form surfaces
T. Nguyen, J. Peters (UF)
GMP 2016
14 / 20
Applications
Applications: Poisson local refinement
Poisson’s equation on the square [0, 6]2
.
Error -0.08:0.08,
T. Nguyen, J. Peters (UF)
-0.02:0.02,
GMP 2016
-0.01:0.01
15 / 20
Applications
Applications: Poisson
Error and approximate convergence rate (a.c.r.): close to 2−4 ` 1 2 3 4 5
||u − uh ||L2 0.0625 0.0098 9.7e-04 7.17e-5 5.29e-06
T. Nguyen, J. Peters (UF)
a.c.r. ||.||L2 6.4 10.1 13.5 13.56
||u − uh ||L∞ 0.0826 0.0194 0.0015 1.3e-4 9.87e-06
GMP 2016
a.c.r. ||.||L∞ 4.3 12.9 11.5 13.1
||u − uh ||H 1 0.4568 0.1448 0.0225 0.0033 4.78e-04
a.c.r.
3 6 6 6
16 / 20
Applications
Applications: Thin shell Scordelis-Lo roof
T. Nguyen, J. Peters (UF)
GMP 2016
17 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
I
Degrees of freedom = as for PHT splines (C 1 bi-3 spline)
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
I
Degrees of freedom = as for PHT splines (C 1 bi-3 spline)
I
, isogeometric analysis
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
I
Degrees of freedom = as for PHT splines (C 1 bi-3 spline)
I I
, isogeometric analysis / shape
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
I
Degrees of freedom = as for PHT splines (C 1 bi-3 spline)
I I
, isogeometric analysis / shape Thank You & Questions?
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
Summary I
Irregularities in a C 1 bi-3 spline surface
I
Refinable (nested)
I
Degrees of freedom = as for PHT splines (C 1 bi-3 spline)
I I
, isogeometric analysis / shape Thank You & Questions?
T. Nguyen, J. Peters (UF)
GMP 2016
18 / 20
Applications
T. Nguyen, J. Peters (UF)
GMP 2016
19 / 20
Applications
linear independence
` Nonzero BB coefficients • of the basis function f21 . The coefficient marked ` k k additionally with an × is nonzero only for f21 . It is zero for f12 or f11 , k k = 0, . . . , n − 1 and for f21 , k 6= `.
T. Nguyen, J. Peters (UF)
GMP 2016
20 / 20
