Quadrature on the positive real line with quasi and pseudo - PowerPoint PPT Presentation

Gauss-type Quadrature ORF with a parameter Computation Quadrature on the positive real line with quasi and pseudo orthogonal rational functions Adhemar Bultheel (joint work with P. Gonz´ alez-Vera, E. Hendriksen, O. Nj˚ astad) Department of Computer Science K.U.Leuven SC2011 International Conference on Scientific Computing, S. Margherita di Pula, Sardinia, Italy October 10-14, 2011. Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature ORF with a parameter Computation Happy birthday Claude and Sebastiano Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature ORF with a parameter Computation Happy birthday Claude and Sebastiano Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature ORF with a parameter Computation Survey Quadrature on the positive real line and OP ORF (incl. OP /OLP) Quasi and Pseudo versions Quadrature with QORF and PORF Differences and similarities Numerical examples Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Gauss-type quadrature � ∞ Consider integrals I µ ( f ) = 0 f ( x ) d µ ( x ), µ > 0. � ∞ Gauss-type QF: inner product � f , g � µ = 0 f ( x ) g ( x ) d µ ( x ) Construct OP { ϕ n } : ϕ n ⊥ L n − 1 = Π n − 1 Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating polynomial for f in these nodes ⇒ weights { λ k , n > 0 } n k =1 n � I µ ( f ) ≈ I µ n ( f ) = λ k , n f ( x k , n ) . k =1 Equality for f ∈ L n · L n − 1 = L 2 n − 1 = Π 2 n − 1 Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Gauss-type quadrature � ∞ Consider integrals I µ ( f ) = 0 f ( x ) d µ ( x ), µ > 0. � ∞ Gauss-type QF: inner product � f , g � µ = 0 f ( x ) g ( x ) d µ ( x ) Construct OP { ϕ n } : ϕ n ⊥ L n − 1 = Π n − 1 Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating polynomial for f in these nodes ⇒ weights { λ k , n > 0 } n k =1 n � I µ ( f ) ≈ I µ n ( f ) = λ k , n f ( x k , n ) . k =1 Equality for f ∈ L n · L n − 1 = L 2 n − 1 = Π 2 n − 1 Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Gauss-type quadrature � ∞ Consider integrals I µ ( f ) = 0 f ( x ) d µ ( x ), µ > 0. � ∞ Gauss-type QF: inner product � f , g � µ = 0 f ( x ) g ( x ) d µ ( x ) Construct OP { ϕ n } : ϕ n ⊥ L n − 1 = Π n − 1 Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating polynomial for f in these nodes ⇒ weights { λ k , n > 0 } n k =1 n � I µ ( f ) ≈ I µ n ( f ) = λ k , n f ( x k , n ) . k =1 Equality for f ∈ L n · L n − 1 = L 2 n − 1 = Π 2 n − 1 Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Gauss-type quadrature � ∞ Consider integrals I µ ( f ) = 0 f ( x ) d µ ( x ), µ > 0. � ∞ Gauss-type QF: inner product � f , g � µ = 0 f ( x ) g ( x ) d µ ( x ) Construct OP { ϕ n } : ϕ n ⊥ L n − 1 = Π n − 1 Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating polynomial for f in these nodes ⇒ weights { λ k , n > 0 } n k =1 n � I µ ( f ) ≈ I µ n ( f ) = λ k , n f ( x k , n ) . k =1 Equality for f ∈ L n · L n − 1 = L 2 n − 1 = Π 2 n − 1 Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Gauss-type quadrature � ∞ Consider integrals I µ ( f ) = 0 f ( x ) d µ ( x ), µ > 0. � ∞ Gauss-type QF: inner product � f , g � µ = 0 f ( x ) g ( x ) d µ ( x ) Construct OP { ϕ n } : ϕ n ⊥ L n − 1 = Π n − 1 Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating polynomial for f in these nodes ⇒ weights { λ k , n > 0 } n k =1 n � I µ ( f ) ≈ I µ n ( f ) = λ k , n f ( x k , n ) . k =1 Equality for f ∈ L n · L n − 1 = L 2 n − 1 = Π 2 n − 1 Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Gauss-type quadrature � ∞ Consider integrals I µ ( f ) = 0 f ( x ) d µ ( x ), µ > 0. � ∞ Gauss-type QF: inner product � f , g � µ = 0 f ( x ) g ( x ) d µ ( x ) Construct OP { ϕ n } : ϕ n ⊥ L n − 1 = Π n − 1 Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating polynomial for f in these nodes ⇒ weights { λ k , n > 0 } n k =1 n � I µ ( f ) ≈ I µ n ( f ) = λ k , n f ( x k , n ) . k =1 Equality for f ∈ L n · L n − 1 = L 2 n − 1 = Π 2 n − 1 Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Gauss-type quadrature � ∞ Consider integrals I µ ( f ) = 0 f ( x ) d µ ( x ), µ > 0. � ∞ Gauss-type QF: inner product � f , g � µ = 0 f ( x ) g ( x ) d µ ( x ) Construct OP { ϕ n } : ϕ n ⊥ L n − 1 = Π n − 1 Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating polynomial for f in these nodes ⇒ weights { λ k , n > 0 } n k =1 n � I µ ( f ) ≈ I µ n ( f ) = λ k , n f ( x k , n ) . k =1 Equality for f ∈ L n · L n − 1 = L 2 n − 1 = Π 2 n − 1 Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Orthogonal Rational Functions Introduce poles: define L n = { f n / d n : f n ∈ Π n } � ζ k − x , if − ∞ < ζ k ≤ 0 d n = r 1 r 2 · · · r n , r k ( x ) = 1 , if ζ k = −∞ Orthogonalize ϕ n ⊥ L n − 1 OP = all ζ ’s at −∞ OLP = all ζ ’s in { 0 , −∞} Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating RF for f in zeros ϕ n ⇒ I µ ( f ) ≈ I µ n ( f ) = � n k =1 λ k , n f ( x k , n ) (equality for f ∈ L n · L n − 1 � = ? L 2 n − 1 ) Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Orthogonal Rational Functions Introduce poles: define L n = { f n / d n : f n ∈ Π n } � ζ k − x , if − ∞ < ζ k ≤ 0 d n = r 1 r 2 · · · r n , r k ( x ) = 1 , if ζ k = −∞ Orthogonalize ϕ n ⊥ L n − 1 OP = all ζ ’s at −∞ OLP = all ζ ’s in { 0 , −∞} Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating RF for f in zeros ϕ n ⇒ I µ ( f ) ≈ I µ n ( f ) = � n k =1 λ k , n f ( x k , n ) (equality for f ∈ L n · L n − 1 � = ? L 2 n − 1 ) Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Orthogonal Rational Functions Introduce poles: define L n = { f n / d n : f n ∈ Π n } � ζ k − x , if − ∞ < ζ k ≤ 0 d n = r 1 r 2 · · · r n , r k ( x ) = 1 , if ζ k = −∞ Orthogonalize ϕ n ⊥ L n − 1 OP = all ζ ’s at −∞ OLP = all ζ ’s in { 0 , −∞} Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating RF for f in zeros ϕ n ⇒ I µ ( f ) ≈ I µ n ( f ) = � n k =1 λ k , n f ( x k , n ) (equality for f ∈ L n · L n − 1 � = ? L 2 n − 1 ) Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Orthogonal Rational Functions Introduce poles: define L n = { f n / d n : f n ∈ Π n } � ζ k − x , if − ∞ < ζ k ≤ 0 d n = r 1 r 2 · · · r n , r k ( x ) = 1 , if ζ k = −∞ Orthogonalize ϕ n ⊥ L n − 1 OP = all ζ ’s at −∞ OLP = all ζ ’s in { 0 , −∞} Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating RF for f in zeros ϕ n ⇒ I µ ( f ) ≈ I µ n ( f ) = � n k =1 λ k , n f ( x k , n ) (equality for f ∈ L n · L n − 1 � = ? L 2 n − 1 ) Adhemar Bultheel Quasi and pseudo ORF for QF on R +

Gauss-type Quadrature Orthogonal polynomials ORF with a parameter Orthogonal Rational Functions (ORF) Computation Orthogonal Rational Functions Introduce poles: define L n = { f n / d n : f n ∈ Π n } � ζ k − x , if − ∞ < ζ k ≤ 0 d n = r 1 r 2 · · · r n , r k ( x ) = 1 , if ζ k = −∞ Orthogonalize ϕ n ⊥ L n − 1 OP = all ζ ’s at −∞ OLP = all ζ ’s in { 0 , −∞} Zeros ϕ n : { x k , n } n k =1 ⊂ R + ⇒ nodes Interpolating RF for f in zeros ϕ n ⇒ I µ ( f ) ≈ I µ n ( f ) = � n k =1 λ k , n f ( x k , n ) (equality for f ∈ L n · L n − 1 � = ? L 2 n − 1 ) Adhemar Bultheel Quasi and pseudo ORF for QF on R +